A TOPOLOGICAL STUDY OF TOROIDAL DYNAMICS

By

Hitesh Gakhar

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Mathematics – Doctor of Philosophy

2020

ABSTRACT

A TOPOLOGICAL STUDY OF TOROIDAL DYNAMICS

By

Hitesh Gakhar

This dissertation focuses on developing theoretical tools in the field of Topological Data
Analysis and more specifically, in the study of toroidal dynamical systems. We make con-
tributions to the development of persistent homology by proving Künneth-type theorems,
to topological time series analysis by further developing the theory of sliding window em-
beddings, and to multiscale data coordinatization in topological spaces by proving stability
theorems.

First, in classical algebraic topology, the Künneth theorem relates the homology of two
topological spaces with that of their product. We prove Künneth theorems for the persistent
homology of the categorical and tensor product of filtered spaces. That is, we describe the
persistent homology of these product filtrations in terms of that of the filtered components.
Using these theorems, we also develop novel methods for algorithmic and abstract compu-
tations of persistent homology. One of the direct applications of these results is the abstract
computation of Rips persistent homology of the N-dimensional torus.

Next, we develop the general theory of sliding window embeddings of quasiperiodic
functions and their persistent homology. We show that the sliding window embeddings
of quasiperiodic functions, under appropriate choices of the embedding dimension and time
delay, are dense in higher dimensional tori. We also explicitly provide methods to choose
these parameters. Furthermore, we prove lower bounds on Rips persistent homology of these
embeddings. Using one of the persistent Künneth formulae, we provide an alternate algo-
rithm to compute the Rips persistent homology of the sliding window embedding, which
outperforms the traditional methods of landmark sampling in both accuracy and time. We
also apply our theory to music, where using sliding windows and persistent homology, we

characterize dissonant sounds as quasiperiodic in nature.

Finally, we prove stability results for sparse multiscale circular coordinates. These coor-
dinates on a data set were first created to aid non-linear dimensionality reduction analysis.
The algorithm identifies a significant integer persistent cohomology class in the Rips filtra-
tion on a landmark set and solves a linear least squares optimization problem to construct a
circled valued function on the data set. However, these coordinates depend on the choice of
the landmarks. We show that these coordinates are stable under Wasserstein noise on the
landmark set.

To my family & friends

iv

ACKNOWLEDGEMENTS

The last few years of my mathematical career, as cliché as it may sound, have been full of ups
and downs. I would not have been able to make it to the finish line if not for the wonderful
people I have met.

First and foremost, I am thankful for my advisor Dr. Jose Perea. Moving from pursuing
pure topology to applied topology was one of the hardest decisions I had to make, and if
it was not for his guidance, expertise, and well chosen problems, this transition would have
been a much harder one. His wisdom and work ethic continues to inspire me. I would also
like to express the deepest of gratitude towards the members of my dissertation committee:
Dr. Matthew Hedden, Dr. Elizabeth Munch, and Dr. Guowei Wei for their guidance through
different stages of my Ph.D. career.

There are two people who facilitated two major plot twists in my life. First, it was
Bhavya Aggarwal who unintentionally made me question my career choices towards the end
of high school which led me to switching from engineering to mathematics. Second, it was
Harrison LeFrois who gave me an initial nudge towards considering applied topology as a
field of research for my dissertation.

My education in mathematics and science started at the Indian Institute of Science
I am thankful for some of the faculty members who
Education and Research, Mohali.
contributed immensely to building my scientific temper, mathematical rigor, and academic
integrity: Dr. Lingaraj Sahu, Dr. Kapil Paranjape, Dr. Amit Kulshrestha, Dr. Krishnendu
Gongopadhyay, Dr. Lolitika Mandal, and Dr. N.G. Prasad.

After moving to Michigan State University, I took many advanced courses in mathematics
to expand my knowledge base. Thank you to all the faculty members who taught me in the
last few years, with a special mention to Dr. Teena Gerhardt for teaching the qualifying
course in introductory algebraic topology.

I have been very fortunate to have access to the teaching resources and opportunities at

v

Michigan State. The encouragement to explore different styles of teaching that I received
from the Central for Instructional Mentoring, in particular, Dr. Tsveta Sendova and Andy
Krause, allowed me to grow as an educator. I am also grateful for Tsveta’s open door policy,
which I utilized at every opportunity, be it a crisis or a moment of celebration. Without her
presence in Wells Hall, this journey would have been arduous. I am glad to have worked
on teaching assignments with the following people: Dr. Peter Magyar, Jane Zimmerman,
Andrew Krause, Sue Allen, Dr. Russell Schwab, and Dr. Shiv Karunakaran. I am thankful
for every feedback, critique, compliment, and opportunity that has been given to me.

I would also like to thank the administrative staff in the mathematics department for
all their work, especially Estrella Starn for her emails to the graduate students during the
COVID-19 crisis, and Ami McMurphy and Nicole Holton for always stopping me in my
tracks and genuinely asking me how I was doing.

On the mathematical side, I would first like to thank my collaborators Dr. Joshua Mike,
Luis Polanco, and Joseph Melby.
It has been a pleasure working with them on different
projects and I am extremely grateful for their assistance with both the theoretical and com-
putational aspects of research. I am also thankful to Ákos Nagy for useful academic advice
over the years. Having a large Geometry and Topology student group in the department has
been very beneficial as it led to many productive discussions, seminar talks, and fun graduate
conferences with Sanjay Kumar, Abhishek Mallick, Wenzhao Chen, Gorapada Bera, Brandon
Bavier, Keshav Sutrave, Danika Van Niel, and Chloe Lewis. I am also happy to have known
Michael Schultz, Mollee Shultz, Sami Merhi, Sugil Lee, Nick Ovenhouse, Dimitris Vardakis,
Ioannis Zachos, Rami Fakhry, Rodrigo Matos, Leonardo Abbrescia, Jihye Hwang, Valentin
Kuechle, Lubhashan Pathirana, Chamila Malagoda, Armstrong Guan, Nick Rekuski, and
Arman Tavakoli during my time at Wells Hall.

Outside the department, I have had the pleasure of meeting many wonderful people. I
am very happy to have known my Spartan Village gang: Akshit Sarpal, Apratim Mukherjee,
Sanjana Mukherjee, and Salil Sapre for being a crucial part of my first family in this country.

vi

While most of them graduated, Salil stayed and indirectly introduced me to more awesome
folks: Charuta Parkhi, Amit Sharma, Manasi Mishra, Hetal Adhia, and most importantly
Hariharan Iyer, who I have enjoyed watching Gunda with on numerous occasions.

I am thankful to Ronak Sripal for introducing me to most of the Desi Boys: Abhiroop
Ghosh, Bharath Basti Shenoy, Sai Chaitanya, Ankit Gautam, Abu Bakr, Siddharth Shukla,
Karthik Krishnan, Anirudh Bhat, and Yash Roy. My weekly nights with them over the last
couple years were a great way to unwind, whether it was playing pool, or getting dinner and
drinks, watching a movie, playing board games, or Hookah at unexpected times. I have also
cherished my time that I have spent with Hattie Cable, Sophia Zhang, Veditha Shetty, Ajay
Dhara, and Anna Skelton.

There are a few people who believed in me and cheered me on during my Ph.D.: Priyanka
Jamadagni, Kritika Singhal, Sujoy Mukherjee, Rhea Palak Bakshi, Bharat Gehlot, Indrajeet
Sagar, Nishika Bhatia, Ben Mackey, Lauren Conway, Stevie Simmons, and Ayesha Yala-
marthy. Thank you for all your support!
I am also grateful to Christos Grigoriadis for
teaching me expletives in Greek, Eylem Yildiz for talking about politics over Indian food,
and Allison Glasson for the wackiest of surprises and telling me, “I believe in you”.

A set of wonderful friends, who I (secretly) call my Review Crew, read through tons of
my abstracts, cover letters, reports, and statements. They helped me turn my drafts into
something of value. In particular, I am thankful to:

• Sarah Klanderman, for multiple discussions on homotopy colimits, long mid-day coffee

walks, and being my most loyal writing partner.

• Charlotte Ure, for countless conversations, inspiring me through her badassery, and

being the best office mate I could hope for.

• Tyler Bongers, for answering my analysis related questions, many adventures at Pinball

Pete’s, getting me into craft beers, and sharing the love of terrible puns.

vii

• Rani Satyam, for Bollywood movie nights, Pokewalks, listening to my pointless venting
about everything and everyone, and being my work accountability buddy. I owe my
morning schedule to our CVS meetups.

• Reshma Menon, for years of soul-saving Indian food, working together in GSC, and
exchanging funny and sad stories about dating mishaps. We started our Ph.D. journeys
together, got our first jobs on the same day, and shared most of the struggles in between.
You were my Ph.D. soulmate.

My partner, Livy Drexler is one of the kindest people I know. The last year of my
dissertation was probably the hardest and the busiest stretch of time in my entire life. Her
companionship has been one of the most important things that has kept me afloat. I am
thankful for it along with the joy and optimism that she brings to my life. I am also thankful
to her for introducing me to her wonderful parents: John and Amy Drexler, who treat me
as a member of their family, and more importantly, send me fine chocolates throughout the
year, and to her dog, Pixie, who makes my life less mundane.

Finally, and most importantly, I am eternally grateful to my parents, Ashok and Usha,
for everything. It is only because of you that I have made it this far in life. Not only did
you provide for me the best education possible, you lent me your unwavering support. The
amount of flexibility and freedom I had in regards to choosing my career path was significant.
Thank you for doing the right thing before 3 Idiots was released. I would also like to thank
my brother, Nivesh, for always having my back no matter what. It has meant a lot to me.

viii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER 1

1.1.1 Persistent Künneth theorems

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Outline & Overview of the dissertation . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
1.1.1.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . .
Sliding window embeddings of quasiperiodic functions . . . . . . . . .
1.1.2.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . .
Stability of circular coordinates . . . . . . . . . . . . . . . . . . . . .
1.1.3.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.2

1.1.3

CHAPTER 2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Category Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Definitions and Examples
. . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Functors and natural transformations . . . . . . . . . . . . . . . . . .
2.1.3 Coequalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Algebraic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 The Classical Künneth Theorems . . . . . . . . . . . . . . . . . . . .
2.2.2 Homotopy Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Principal bundles and Čech cohomology . . . . . . . . . . . . . . . .
2.3 Persistent Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Takens’s embedding theorem . . . . . . . . . . . . . . . . . . . . . . .
2.5 Analytical tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Kronecker’s theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 Wasserstein metric on probability measures
. . . . . . . . . . . . . .

CHAPTER 3 PERSISTENT KÜNNETH THEOREMS . . . . . . . . . . . . . . .
3.1 Products of Diagrams of Spaces . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 A comparison theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A Persistent Künneth formula for the categorical product . . . . . . . . . . .
3.3 A Persistent Künneth Formula for the generalized tensor product
. . . . . .
3.3.1 The tensor product and Tor of graded modules
. . . . . . . . . . . .
3.3.2 The Graded Algebraic Künneth Formula . . . . . . . . . . . . . . . .
3.3.3 A Persistent Eilenberg-Zilber theorem . . . . . . . . . . . . . . . . . .
3.3.4 Barcode Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Application: Persistent Homology of N-torus . . . . . . . . . . . . . . . . . .

CHAPTER 4 QUASIPERIODICITY . . . . . . . . . . . . . . . . . . . . . . . . .

ix

1
3
3
6
7
10
11
14

15
15
15
17
19
20
20
22
25
31
39
41
46
46
47
48

50
50
58
61
64
65
68
70
77
79

81

4.0.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Quasiperiodic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
“Fourier series” of a quasiperiodic function . . . . . . . . . . . . . . . . . . .
4.2
4.2.1 The iterated Fourier series . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Analyzing the coefficients ˆF(k)
. . . . . . . . . . . . . . . . . . . . .
4.2.3 Approximation theorems for sliding window embeddings
. . . . . . .
. . . . . . . . . . . . . . . . . . . . .

81
81
84
84
88
93
4.3 The geometric structure of SWd,τ SZ f
95
4.4 Persistent Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5 Computational Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.5.1 An approximation strategy using a persistent Künneth formula . . . . 110
. . . . . . . . . . . . . . . . 111
4.5.1.1 An example with 2 frequencies
4.5.1.2 Computational Results . . . . . . . . . . . . . . . . . . . . . 112
4.5.2 An application to music theory . . . . . . . . . . . . . . . . . . . . . 114

CHAPTER 5 STABILITY OF MULTISCALE CIRCULAR COORDINATES . . . . 117
5.0.1 The sparse circular coordinates algorithm . . . . . . . . . . . . . . . . 117
5.1 The stability set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1.1 Partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.2 Weights on edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
. . . . . . . . . . . . . . . . . . . . . . 130
5.3.1 Landmark Augmentation . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3.2 Coordinates old and new . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.3.3 Wasserstein stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Example & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.2 Geometric noise model
5.3 The general Wasserstein noise model

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

x

LIST OF TABLES

Table 4.1: Example of τ estimation: Candidates for τ, along with the corresponding

local minimums, and the angles θk,l (in degrees) between uk and ul

. . . 101

Table 4.2: Computational times, and number of points, for the landmark and Kün-

neth approximations to bcdR

i (SWd,τ f), i ≤ 2. . . . . . . . . . . . . . . . . 113

Table 4.3: Areas for confidence regions from bcdR

i (XL × YL)
(red), i = 1, 2 (see Figure 4.4). A smaller area suggests a better approx-
imation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

i (L) (blue) and bcdR

Table 4.4: List of frequencies in the amplitude-frequency spectrum: The first row
is the list of frequencies that were detected. The second row is their
conversion to Hertz. The third row is their ratio with respect to the first one.115

xi

LIST OF FIGURES

Figure 1.1: A filtered simplicial circle X . The numbers next to simplex indicate

when that corresponding simplex appeared in the filtration.

. . . . . . .

Figure 1.2: The product filtrations X × X (left) and X ⊗ X (right). The numbers
next to simplex indicate when that corresponding simplex appeared in
the filtration.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1.3:

In the leftmost figure, we see a quasiperiodic function.
In the mid-
dle figure, we see a PCA visualization of the reconstructed torus from
sliding window embedding. In the rightmost figure, we see the 1 and
2-dimensional Rips persistence diagrams (blue and red points respec-
tively) of the sliding window point cloud. . . . . . . . . . . . . . . . . . .

Figure 1.4: (Left) The data set X of 1000 points sampled around a circle, with two
different sets of 40 landmarks (Middle). For each of these sets, we obtain
a function that assigns to each data point in X a circular coordinate.
(Right) We see a comparison of the two coordinate functions.
Figure 2.1: A filtered simplicial circle K = {Ki}i, with Ki = K5 for i ≥ 5.
Figure 2.2: The Lorenz attractor for parameters σ = 10, β = 8
3, and ρ = 28.

. . . . . .

. . . . . .

. . . . .

Figure 2.3: The evolution of (0, 0) under the irrational flow φ on T2 as defined in

Example 2.4.2 for ω =

3. . . . . . . . . . . . . . . . . . . . . . . . . . .

√

Figure 2.4: Time series associated to the observation f of the evolution of (0, 0) for

the irrational flow φ.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

5

10

13

37

39

40

42

Figure 2.5: Dynamical System to Sliding Window Reconstruction for a counter-
clockwise flow on S1: In the leftmost figure, we see a standard circle.
In the middle, we see the time series generated g(t). In the rightmost
figure, we see the reconstructed circle from sliding window embedding.

.

44

Figure 2.6: Dynamical System to Persistence for a rational flow on T2: In the left-
most figure, we see a rational flow on T2.
In the second, we see the
time series generated g(t). In the third figure, we see a PCA visualiza-
tion of the reconstructed circle from sliding window embedding. In the
rightmost figure, we see the 1- dimensional Rips persistence diagram of
a sliding window point cloud sampled from the third figure.

. . . . . . .

44

xii

Figure 2.7: Dynamical System to Persistence for an irrational flow on T2: In the
leftmost figure, we see an irrational flow on T2. In the second, we see
the time series generated g(t). In the third figure, we see a PCA visu-
alization of the reconstructed torus from sliding window embedding. In
the rightmost figure, we see the Rips persistence diagrams in dimensions
1, 2 of a sliding window point cloud sampled from the third figure. . . . .

Figure 3.1: Visual representation for the deformation retract H3.
√
√
3) ≡ (
Figure 4.1: The sets Zβ for tuples β = (
2,
√
√
√
√
3} (third), and β = (
2,
2 + 2
3, 3

. . . . . . . . . . .
√
√
3, 0) (left), β = (
2+
√
3, π) (right).

3) (second), {√
2,

√
2,

√

√

√

2,

3,

45

58

98

Figure 4.2: Example: The objective function G(x) for finding an optimal τ, for the

√
function f(t) = eit + ei

Figure 4.3: (Left) real(cid:16)

f(t)(cid:17) = 1√

√
3t + ei(
2 cos(t) + 1√

3+1)t.

2 cos(

. . . . . . . . . . . . . . . . . . . 101
√
3t) for the quasiperiodic time

series f, and (Right) a PCA projection into R3 of the point cloud SWd,τ f. 111

Figure 4.4: Confidence regions for bcdR

i (XL × YL), i = 1, 2. For
a given color, each box (confidence region) contains a point (‘J , ρJ)
corresponding to a unique J ∈ bcdR

i (L) and bcdR

Figure 4.5: The audioread plot of a dissonant music sample created on a brass horn.

i (SWd,τ f). . . . . . . . . . . . . . . . 114
115

Figure 4.6: The amplitude-frequency spectrum of the music sample plot using Fast

Fourier Transform.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Figure 4.7: (Left) PCA representation of the sliding window point cloud generated
from the dissonant sample with parameters d = 9 and τ = 0.02858.
(Right) Persistence Diagrams in homological dimension 1 and 2. The
dotted lines separate the top 2 persistence points (sim. maximum per-
sistence point) from the rest in dimension 1 (sim. dimension 2).

. . . . . 116

Figure 5.1: An example of reconstructed toroidal dynamics: (Left) The irrational
√
3. (Second) The
winding on a 2-torus starting at (0, 0) with slope
sliding window embedding of the observation f(t) = 1√
2 eit + 1√
3t of
the winding. (Third) The Rips persistence diagram in dimension 0, 1.
The two 1-persistent cocycles can be used to create two sets of circular
coordinates. (Right) The pair of coordinates can be used to reconstruct
the toroidal winding. The average slope is estimated to be 1.9698. . . . . 119

2 ei

√

xiii

Figure 5.2: An example demonstrating stability of Circular Coordinates: (Left) The
data set X of 1000 points sampled around a circle, with two different sets
of 40 landmarks (Middle). For each of these sets, we obtain a function
that assigns to each data point in X a circular coordinate. (Right) We
see a comparison of the two coordinate functions.

. . . . . . . . . . . . . 145

Figure 5.3: Comparison between Hausdorff and Wasserstein noise: (Left) Example
of the dataset in black and 16 landmarks shown in blue. (Right) Com-
parison of the number of replicates against the incurred error between
the circular coordinates. Red shows the initial ‘Zero-Hausdorff-distance’
pair, while green shows the second ‘Zero-Wasserstein-distance’ pair.

. . 146

xiv

CHAPTER 1

INTRODUCTION

The story begins with the need to understand time series data. Such data prevails in this
world and makes for an important object to study. As a result, there are many methods to
study it. Some of these include the Discrete Fourier Transform, Wavelet analysis, correlation
methods, etc. One method for detecting recurrence in time series data was recently developed
[59] using ideas from geometry and topology. This novel framework used two key ingredients,
sliding window embeddings and persistent homology.

The first of these ingredients, sliding window embeddings, were originally used in the study
of dynamical systems to reconstruct the topology of underlying dynamics, under smoothness
conditions given by Takens [67], from generic observation functions. It was shown in [59]
that the sliding window embedding of a periodic function is homeomorphic to a circle. We
study a closely related class of functions, namely quasiperiodic functions, which are defined
as a superposition of periodic functions with non-commensurate harmonics. Such dynamics
appear naturally in biphonation phenomena in mammals [74], as well as in the transition to
chaos in rotating fluids [37].

An important point conveyed by Takens’ theorem [67] is that the topological shape of
the sliding window embedding can be measured and leveraged for applications. To measure
this shape, we turn to our second ingredient, persistent homology [28, 76]. It is an algebraic
and computational tool widely used in Topological Data Analysis (TDA) to quantify mul-
tiscale features of shapes. The input to a persistent homology computation is a sequence
of topological spaces called a filtration, while the output is a multiset of intervals called a
barcode, which constitutes a finite topological invariant of the coarse geometry governing the
shape of a given point cloud.

Some of its applications to science and engineering include recurrence detection in time
series data [53, 57, 59], coverage problems in sensor networks [23], the identification of spaces

1

of fundamental features from natural scenes [15], inferring spatial properties of unknown
environments via biobots [26], and many more. Some of the underlying ideas of persistence
are also starting to permeate pure mathematics, most notably symplectic geometry [61, 70].
Although persistent homology is informative about the topology of data sets, one can
further use it to perform nonlinear dimensionality reduction, for example on the point cloud
obtained from sliding window embeddings, for the purposes of visualization and reconstruc-
tion of the underlying dynamics. This recoordinatization process, called Eilenberg-MacLane
coordinatization, turns persistent cohomology classes into maps from point clouds into vari-
ous Eilenberg-MacLane spaces, like the circle [25, 55], real or complex projective space [54],
or lens spaces [60].

This dissertation focuses on developing theoretical tools in the field of TDA and more
specifically, in the study of dynamical systems on tori. There are three major contributions:

1. Theoretical development of persistent homology: We develop two persistent
Künneth formulae, that is, results relating persistent homology of two filtered spaces
to persistent homology of their products. Leveraging these formulae, we present novel
methods for algorithmic and abstract computations of persistent homology.

2. Theoretical foundations of sliding window embeddings: We prove foundational
results in the theory of sliding window embeddings for quasiperiodic functions and
develop a theoretical framework to detect quasiperiodicity in time series data using
sliding window embeddings and a persistent Künneth formula for Rips complexes.

3. Stability results for circular coordinates: The circular coordinatization algorithm
identifies a significant Rips persistent cohomology class on a subset of the point cloud
called the landmarks, and solves a linear least squares optimization problem to con-
struct a circle valued function on the data set. We prove that circular coordinates are
stable under Wasserstein noise on the landmark set.

2

1.1 Outline & Overview of the dissertation

Due to the diverse nature of research problems discussed here, a wide range of mathe-
matical concepts were used. While a comprehensive review for all of them is beyond the
scope of this dissertation, we provide the necessary background in Chapter 2. We prove
the two persistent Künneth formulae in Chapter 3 and revisit the persistent homology of an
N-torus as an application. In Chapter 4, we develop the theory of sliding window embed-
dings for quasiperiodic functions. Using theorems from Chapter 3, we provide an alternate
strategy to recover toroidal features from the sliding window point clouds. We also present
an application where we characterize dissonance in music theory as a quasiperiodic feature.
In Chapter 5, we prove stability results for the construction of sparse circular coordinates
algorithm under Wasserstein noise on the landmarks. Next we provide an overview of the
main results in this dissertation.

1.1.1 Persistent Künneth theorems

For a Principal Ideal Domain (PID) R and topological spaces X and Y , the classical Künneth
formula given by the split natural short exact sequence

0 →M

(Hi(X) ⊗R Hn−i(Y )) → Hn(X × Y ) →M

(TorR(Hi(X), Hn−i−1(Y ))) → 0 (1.1)

i

i

relates the homology of the product space X × Y to the homology of spaces X and Y , with
coefficients in R. As discussed above, the persistent homology algorithm takes a sequence of
topological spaces or simplicial complexes, called a filtration

X : X0 ,→ X1 ,→ X2 ,→ ··· ,→ Xd ,→ ···

where d ∈ N0 = N ∪ {0}, as the input.
It returns in homological dimension n, a bar-
code bcdn(X ) which is a collection of real intervals, or equivalently a persistence diagram
dgmn(X ), which is a multiset of points in the plane R2. In this dissertation, we exploit the
equivalence of these two invariants and use them interchangeably. A brief review is provided

3

in section 2.3.
In Chapter 3, we develop Künneth formulae for persistent homology. For filtered topological
spaces X and Y, we consider two notions of product filtrations:

(X ⊗ Y)d = [

i+j=d

(X × Y)d = Xd × Yd

and

Xi × Yj.

These choices are motivated by different observations. The first choice of filtration, which
we refer to as the categorical product, is of interest for two reasons: (a) it is the product in
the category of filtered topological spaces, and (b) it is relevant in certain computational
settings. More precisely, if (X, dX) is a metric space and

R
(X, dX) = {σ ⊂ X : |σ| < ∞, diam(σ) ≤ 
}

is its Rips complex at parameter 
, then R
(X × Y, dX×Y ) = R
(X, dX) × R
(Y, dY ) [1,
Proposition 10.2]. Here dX×Y is the maximum metric

(cid:16)(x, y), (x0, y0)(cid:17) = max{dX(x, x0), dY (y, y0)},

dX×Y

and the product of Rips complexes is in the category of abstract simplicial complexes (see
section 3.1). The other choice of filtration, which we call the tensor product, is relevant due
to an algebraic resemblance between the short exact sequence it satisfies and the classical
equation 1.1. In Figure 1.1, we see an example of a topological circle X , while in Figure 1.2,
we see two filtrations of a 2-torus defined by our products.

Figure 1.1: A filtered simplicial circle X . The numbers next to simplex indicate when that
corresponding simplex appeared in the filtration.

In the following theorem, we state the two Künneth formulae in terms of their barcode

representation:

4

Figure 1.2: The product filtrations X × X (left) and X ⊗ X (right). The numbers next to
simplex indicate when that corresponding simplex appeared in the filtration.

Theorem 1.1.1. Let X and Y be pointwise finite filtrations. Then the barcodes of the
categorical product X × Y are given by the (disjoint) union of multisets

(1.2)

Similarly, the barcodes for the tensor product filtration X ⊗ Y satisfy

bcdn (X × Y) = [

.

(cid:27)

(cid:26)

i+j=n

I ∩ J

(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ) , J ∈ bcdj(Y)
)
(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ) , J ∈ bcdj(Y)
[
)
(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ) , J ∈ bcdj(Y)
(ρJ + I) ∩ (ρI + J)

(
(‘J + I) ∩ (‘I + J)
(

bcdn(X ⊗ Y) = [
[

i+j=n

i+j=n−1

where ‘J and ρJ denote, respectively, the left and right endpoints of the interval J.

The first Künneth formula implies the following corollary for the Rips persistent homology

(i.e. persistent homology of Rips filtration) of product metric spaces:
Corollary 1.1.2. Let (X, dX), (Y, dY ) be finite metric spaces and let bcdR
n-th dimensional barcode of the Rips filtration R(X, dX) := {R
(X, dX)}
≥0. Then,
(cid:27)
j (Y, dY )

n (X × Y, dX×Y ) = [

i (X, dX) , J ∈ bcdR

(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdR

bcdR

I ∩ J

(cid:26)

i+j=n

n (X, dX) be the

for all n ∈ N0, where dX×Y is the maximum metric.

5

The above formulae also hold true when the inclusions Xi ,→ Xi+1 and Yi ,→ Yi+1 are
replaced by continuous maps. The categorical product of R-indexed diagrams is just the
R-indexed diagram defined using index-wise products. That is,

(X × Y)r = Xr × Yr

for all r ∈ R. The generalized tensor product ⊗g of N0-indexed diagrams —equivalent to
⊗ for inclusions— is defined as follows: the space (X ⊗g Y)k is the homotopy colimit (see
section 2.2.2) of the functor from the poset 4k = {(i, j) ∈ N2 : i + j ≤ k} to Top sending
(i, j) to Xi × Yj, and (i ≤ i0, j ≤ j0) to the map Xi × Yj → Xi0 × Yj0 induced by composite
maps Xi → Xi0 and Yj → Yj0. The map (X ⊗g Y)k → (X ⊗g Y)k+1 is the one induced at
the level of homotopy colimits by the inclusion 4k ⊂ 4k+1.

1.1.1.1 Related work

Different versions of the algebraic Künneth theorem in persistent homology have appeared
in the last three years. In [62, Proposition 2.9], the authors prove a Künneth formula for
the tensor product of filtered chain complexes, while [12, Section 10] establishes Künneth
theorems for the graded tensor product and the sheaf tensor product of persistence modules.
In [14], the authors prove a Künneth formula relating the persistent homology of metric
spaces (X, dX), (Y, dY ) to the persistence of their cartesian product equipped with the L1
(sum) metric (X×Y, dX +dY ), for homological dimensions n = 0, 1. The persistent Künneth
theorems proven in this dissertation and [35] are the first to start at the level of filtered spaces,
identifying the appropriate product filtrations and resulting barcode formulae. Compared
to the sum metric [14], we remark that our results for the maximum metric hold in all
homological dimensions.

6

1.1.2 Sliding window embeddings of quasiperiodic functions

The sliding window embedding of a function f : R → C for a fixed embedding dimension
d ∈ N and time delay τ > 0, at t ∈ R is defined as



f(t)

f(t + τ)

...

f(t + dτ)

 .

SWd,τ f(t) =

This definition is motivated by Takens’ embedding theorem which we cover in section 2.4.1.
In its essence, the theorem provides conditions under which a smooth attractor can be
reconstructed from a sequence of observations made with a generic observation function.
While this reconstructed attractor might have a different geometric shape than the original,
its topology is still preserved.

As discussed at the beginning of this chapter, if the set of harmonics controlling a func-
tion is incommensurate, that is if it is linearly independent over Q, we call the function a
quasiperiodic function. More rigorously, we define f : R → C to be quasiperiodic with a
given set of frequencies {ω1, . . . , ωN} if there is a function F : RN → C, called a parent
function, such that F is periodic in the n-th variable with period 2π
, the restriction of F to
ωn
the diagonal takes the values of f, and N is minimal. See Definition 4.1.1. Assuming that
the parent function is continuously differentiable and using tools from multivariate harmonic
analysis, we can write a Fourier-like series that converges everywhere to F and its restriction
to the diagonal gives us the following series for the quasiperiodic function f:

f(t) = F(t, . . . , t) = X

NQ

ωn
n=1
(2π)N

2π

ω1Z

0

k∈ZN

e−ik1ω1t1

ˆF(k)eithk,ωi
ωNZ
ω2Z

2π

2π

···

0

0

ˆF(k) =

F(t1, . . . , tN)e−ikN ωN tN dtN . . . dt1.

where ω is the frequency vector (ω1, . . . , ωN) and h·,·i is the dot product in RN. As in single
variable Fourier analysis, we can use a truncation parameter Z ∈ N0 to approximate f by

7

the partial sum polynomials SZ f:

ˆF(k)eithk,ωi

SZ f(t) = X
o. Since SZ f(t) → f(t) everywhere as Z → ∞ and since

k∈IN
Z

Z =nk ∈ ZN (cid:12)(cid:12)(cid:12) ||k||∞ ≤ Z

where IN
for fixed parameters d, τ, the map f → SWd,τ f is a bounded linear operator (see Proposition
2.4.8), we conclude that SWd,τ SZ f → SWd,τ f everywhere as Z → ∞. The next theorem
gives us a series of bounds —for different levels of regularity of f— on the proximity of
sliding window embeddings of f to that of its truncated polynomial SZ f. Moreover, the
more regular f is, the faster the convergence of sliding window embeddings will be. Then
by the stability theorem for persistent homology [19], we have the following amalgamation
of theorem 4.2.8 and corollary 4.2.9:

Theorem 1.1.3. Let r ∈ N \ {1}. If F ∈ Cr(cid:16)RN , C(cid:17), then for any bounded T ⊂ R,
i (SWd,τ SZ f(T))(cid:17) ≤ 2dH
SWd,τ f(T), SWd,τ SZ f(T)(cid:17)
(cid:16)
i (SWd,τ f(T)), bcdR
r(cid:16)
Area(SN−1)(cid:17)1/2
1/2 √
 NX
√
!1/2
N
√
ωr−1
2r − 2
min

(cid:16)bcdR

d + 1
Zr−1

  NQ

||RZ(∂r

nF)||2

≤ 2

dB

|ωn|

n=1

n=1

where RZ f := f − SZ f, and dH and dB are the Hausdorff and the Bottleneck distance
respectively (see section 4.4).

The theorem tells us that as the sliding window point cloud SWd,τ SZ f(T) approximates
SWd,τ f(T) in the Hausdorff sense, the Rips persistent homology bcdR
i (SWd,τ SZ f) converges
to bcdR
i (SWd,τ f) in the Bottleneck sense. This suggests that one could replace the study of
a function f with that of its approximation SZ f. Accordingly, we study the geometric struc-
ture of the sliding window embedding of the truncated polynomial SZ f. We see next that
under appropriate conditions on d, τ, and Z, the closure of the sliding window embedding
SWd,τ SZ f is homeomorphic to TN wrapped around Td+1. Indeed, we have the following
structure theorem:

8

Theorem 1.1.4. The sliding window embedding SWd,τ SZ f(Z) = n

SWd,τ SZ f(z) | z ∈ Zo

is dense in a space homeomorphic to TN in Td+1 if

• d + 1 ≥ α := the number of terms in IN

Z with non zero coefficients, and

• τ satisfies that G(τ) < (d + 1)2 for

G(x) = X

(cid:12)(cid:12)(cid:12)1 + eixhk−l,ωi + ··· + eixhk−l,ωid(cid:12)(cid:12)(cid:12)2

.

k6=l∈IN
Z

The condition on τ only guarantees the same topology as a torus. Unfortunately, the
Rips persistent homology cares about the underlying geometry and that requires τ to satisfy
an extra condition: for some practically reasonable large Υ ∈ R+, τ satisfies

τ = argmin
x∈[0,Υ]

G(x).

The last main result we present here pertains to the Rips persistent homology of SWd,τ SZ f.
Let λmin be the square of the minimum singular value of



ΩZ,f =

eihk1,ωiτ0

...

eihk1,ωiτ d

eihkα,ωiτ0

···
...
··· eihkα,ωiτ d,

...



and | ˆF(k)min| be the smallest non-zero Fourier coefficient in absolute value. Then we have
the following theorem:
Theorem 1.1.5. For i ≤ N, there will be at least (cid:16)N
(cid:17) bars
i ))
(0, d1), (0, d2), . . . , (0, d(N
(cid:17).
SWd,τ SZ f,||·||2

where dn ≥ √

3| ˆF(k)min|√

(cid:16)

λmin in bcdR

i

i

We finish this section with an example:

Example 1.1.6. Assume that we are given a quasiperiodic time series f(t) = cos(t) +
√
3t). The sliding window embedding SWd,τ f(R) of f with appropriate dimension and
cos(
delay recovers the topology of T2. Rips persistent homology confirms that in the accompa-
nying figure.

9

Figure 1.3: In the leftmost figure, we see a quasiperiodic function. In the middle figure, we
see a PCA visualization of the reconstructed torus from sliding window embedding. In the
rightmost figure, we see the 1 and 2-dimensional Rips persistence diagrams (blue and red
points respectively) of the sliding window point cloud.

1.1.2.1 Related work

Since the inception of sliding windows and persistence technique for time series analysis in
[59], there have been a few advancements to both theory and applications. The first results
on sliding window embeddings for quasiperiodic functions [53] were restricted to the special
case of sums of exponentials with incommensurate frequencies. Said results are proved in
generality in this dissertation and [36]. While the sliding window embeddings of quasiperiodic
functions trace tori densely, in [75], the authors construct families of time series whose
sliding window embeddings trace Klein bottles, spheres, and projective planes. A recently
published survey ties the worlds of dynamical systems and topological data analysis [57].
Among the applications of sliding window embeddings and persistence, a few are: periodicity
quantification in gene expression data [58]; (quasi)periodicity detection in video data [69];
synthesis of slow-motion videos from recurrent movements [68]; detection of wheeze, which is
an abnormal whistling sound produced while breathing [33]; and detection and classification
of chatter (vibrations caused by machines) in machining [45, 46].

10

1.1.3 Stability of circular coordinates

Circular coordinates on a data set were developed to aid non-linear dimensionality reduction
analysis [25]. Such reduction schemes address the problem of representing high dimensional
data in terms of low dimensional coordinates. The authors used persistent cohomology
to construct circular coordinates on a data set X. The algorithm identifies a significant
integer persistent cohomology class on the Rips filtration and solves a linear least squares
optimization problem to construct a circled valued function on the data set.

Using similar ideas and principal Z-bundles, the sparse circular coordinates were con-
structed by using a landmark set L = {‘1, . . . , ‘N} in lieu of the entire data set X [55].
The idea is as follows. For a fixed prime q > 2, compute the 1-dimensional Rips persistent
cohomology P H1(R(L); Zq). Choose a bar [a, b) from the resultant barcode bcdR(L) sat-
isfying 2a < b. Let 2a < 2α < b and let η00 ∈ Z1(R2α(L); Zq) be a cocycle representative
for the persistent cohomology class corresponding to [a, b). Lift η00 to an integer cocycle
η0 ∈ Z1(R2α(L); Z) such that η00 − η0 mod q is a coboundary in C1(R2α(L); Zq). Lastly,
define η to be the image of η0 under C1(R2α(L); Z) → C1(R2α(L); R) induced by the inclu-
sion ι : Z ,→ R. Choose positive weights for the vertices and edges of R2α(L) and let d+
2α be
the weighted Moore-Penrose pseudoinverse of the coboundary map

and define

d2α : C0(R2α(L); R) → C1(R2α(L); R).

τ := −d+

2α(η)

and

θ := η + d2α(τ).

Denote τj := τ([‘j]) and θjk := θ([‘j, ‘k]) when [‘j, ‘k] ∈ R2α(L), and 0 otherwise. Let
Bα(‘j) denote the ball of radius α centered around ‘j ∈ L, and {ϕj}j a partition of unity
subordinate to the collection {Bα(‘j)}j. The sparse circular coordinates are defined as

11

follows:

hθ,τ :

N[

2πi

τj +

NX

r=1

 .

ϕr(b)θjr

Bα(‘r) → S1 ⊂ C

r=1
Bα(‘j) 3 b 7→ exp

We give a more detailed version of this algorithm in section 5.0.1.

Observe that the coordinates generated from this algorithm depend on the choice of
landmarks. For any two finite landmark sets L, ˜L with probability weights ω and ˜ω, we
obtain two circular coordinate functions h and ˜h respectively. Assume that

dW,∞((L, ω), (˜L, ˜ω)) < δ

where dW,∞ is the Wasserstein-Kantorovich-Rubinstein metric as in Definition 2.5.5. Our
first result shows the stability of circular coordinates for a special case, that is when L and
˜L are in a weight preserving bijective correspondence. We call this the geometric stability of
circular coordinates. The following theorem appears in more detail as theorem 5.2.7.

Theorem 1.1.7. If (L, ω) and (˜L, ˜ω) are landmark sets in a weight preserving bijective
correspondence, and h and ˜h are the associated circle-valued functions, then

||h − ˜h||∞ ≤ Cδ

where C is a term dependent on the choice of weights, the harmonic cocycle, and partitions
of unity used in the construction of circular coordinates.

Let us look at the following example:

Example 1.1.8. We start with a dataset X with 1000 points sampled around a unit circle
with Gaussian noise of 0.5. We first select the “blue” landmark set with 40 points by maxmin
sampling. We define the “red” landmark set by choosing the respective nearest neighbours
of the “blue” landmarks. See Figure 1.4.

12

Figure 1.4: (Left) The data set X of 1000 points sampled around a circle, with two different
sets of 40 landmarks (Middle). For each of these sets, we obtain a function that assigns
to each data point in X a circular coordinate.
(Right) We see a comparison of the two
coordinate functions.

In a more general scenario, L and ˜L might have different cardinalities or vertex probabil-
ity weights. In section 5.3.1, we introduce a way to create two new landmark sets (Lm, ωm),
(˜Lm, ˜ωm) such that they are in a weight preserving bijective correspondence, thereby reduc-
ing the case of Wasserstein stability to the case of geometric stability. Both landmark sets
Lm, ˜Lm are just L, ˜L respectively, augmented with copies of points. The next result de-
scribes the relationship between the circular coordinates generated by L and the augmented
Lm (and similarly for ˜L and ˜Lm).

Theorem 1.1.9. Let Lm be the landmark set augmented from L, and hm and h be the
circular functions associated to them. Then for all b ∈ X, hm(b) = h(b).

See theorem 5.3.13 for more details. As a consequence, we can replace h, ˜h with hm, ˜hm

to obtain

||h − ˜h||∞ = ||hm − ˜hm||∞

where the right hand side can be bounded using bounds the geometric stability theorem.
These arguments are presented in section 5.3.3, in particular summarized in Theorem 5.3.15.

13

1.1.3.1 Related work

In [25], a strategy for non-linear dimensionality reduction was developed using the first (non-
sparse) circular coordinates. In [55], the theory of principal Z-bundles was used to construct
sparse circular coordinates by considering Rips filtration on a landmark set L ⊂ X.
In
[54], the multiscale projective coordinates were constructed, along with a non-linear dimen-
sionality reduction scheme called Principal Projective Component Analysis. Using similar
techniques, lens spaces coordinates were constructed in [60]. The authors also presented
a non-linear dimensionality reduction scheme called LPCA or Lens Principal Component
Analysis.

14

CHAPTER 2

BACKGROUND

This chapter deals with the necessary mathematical background for the later chapters of this
dissertation. We provide a terse introduction to category theory in 2.1. In 2.2, we provide
background material for an assortment of topics in topology. Using ideas from these two
sections, we give a brief background for persistent homology in 2.3. In section 2.4, we discuss
the basics of dynamical systems along with Takens’ theorem and sliding window embeddings.
In section 2.5, we review one variable Fourier series, Kronecker’s approximation theorem, and
Wasserstein metric on probability measures.

2.1 Category Theory

In this section, we give a succinct review of category theory. For a more comprehensive
review, we refer the mathematician reader to [48] and the data scientist reader to [65].
Category theory was originally developed by Eilenberg and MacLane to bridge the gap
between the fields of algebra and topology. More generally, category theory takes a bird eye’s
view of mathematics by considering abstractions of mathematical concepts. The advantage
to this view is that a lot of intricate and subtle results can be proved by ‘cleaner’ methods.
Over the last few decades, category theory has found itself useful in sciences, like quantum
physics[2] and theoretical computer science [3].

2.1.1 Definitions and Examples

Definition 2.1.1. A catgeory C consists of a class of objects denoted by Obj(C) and a set of
morphisms Mor(a, b) for any two objects a, b ∈ Obj(C) such that the following axioms hold:

1. If f ∈ Mor(a, b) and g ∈ Mor(b, c), then g ◦ f ∈ Mor(a, c).

2. For each a ∈ Obj(C), there is a unique morphism 1a ∈ Mor(a, a) called the identity

15

morphism satisfying

f ◦ 1a = f = 1b ◦ f

for any b ∈ Obj(C) and any f ∈ Mor(a, b).

3. For any f ∈ Mor(a, b), g ∈ Mor(b, c), and h ∈ Mor(c, d),

h ◦ (g ◦ f) = (h ◦ g) ◦ f.

Some of the categories that we are interested in are:

Example 2.1.2. Top is an example of a category with topological spaces as objects and
continuous maps as morphisms.

Example 2.1.3. Given a commutative ring R with unity, ModR is the category of all R-
modules as objects and for any two R-modules M, N, the set of morphisms from M to N is
the set of all R-module homomorphisms from M to N.

Example 2.1.4. Given a commutative ring R with unity, ChR is the category of all chain
complexes of R-modules as objects and for any two such chain complexes C, C0, the set of
morphisms from C to C0 is the set of all chain maps from C to C0.

Definition 2.1.5. A category C is small if the class of its objects, Obj(C), forms a set.

In some categories, like Top, the sets of morphisms may be very large in cardinality.
For purposes of this document, we are interested in another special type of category that
restricts the size of morphism classes:

Definition 2.1.6. A category is called thin if there is at most one morphism between any
two objects.

Definition 2.1.7. A subcategory D of C consists of the following data:

1. A subclass Obj(D) of Obj(C),

16

2. For each a, b ∈ Obj(D), a subset of MorC(a, b),
3. For each a ∈ Obj(D), the identity 1a ∈ MorC(a, a) belongs to MorD(a, a), and
4. For each f : MorD(a, b) and g ∈ MorD(b, c), the composition g ◦ f is in MorD(a, c).

Definition 2.1.8. A full subcategory D of C satisfies

MorD(a, b) = MorC(a, b)

for any objects a, b ∈ Obj(D).

In fact, any subclass of Obj(C) yields a unique full subcategory of C.

Definition 2.1.9. For a category C, its opposite category Cop is defined as

Obj(Cop) := Obj(C)

MorCop(a, b) := MorC(b, a),

that is, all the morphisms are reversed.

2.1.2 Functors and natural transformations
Definition 2.1.10. Let C and C0 be categories. A covariant functor D from C to C0
associates to each object, a ∈ Obj(C) an object D(a) ∈ Obj(C0) and to each morphism
f ∈ MorC(a, b), a morphism D(f) ∈ MorC0(D(a), D(b)). Moreover, it is required that the
functor respects morphism composition and preserves the identity morphisms, that is,

D(g ◦ f) = D(g) ◦ D(f) and D(1a) = 1D(a)

respectively.
Similarly, a contravariant functor D from C to C0 associates to each object a ∈ Obj(C),
an object D(a) ∈ Obj(C0) and to each morphism f ∈ MorC(a, b), a morphism D(f) ∈
MorC0(D(b), D(a)). Moreover, it is required that the functor preserves the identity mor-
phisms, however D(g ◦ f) = D(f) ◦ D(g).

17

For example, if C = Top and C0 = ModR, then the functor D that takes a topological
space X to the homology module Hn(X; R) is covariant, while the functor D0 that takes X
to the cohomology module Hn(X; R) is contravariant.
Definition 2.1.11. Let D, D0 be covariant functors from C to C0. A natural transformation
η : D → D0 consists of a family of morphisms η(a) : D(a) → D0(a) for each object a ∈
Obj(C), such that for each morphism f ∈ MorC(a, b), the diagram

η(a)

D(a)

D(f)

D(b)

η(b)

D0(a)

D0(b)

D0(f)

commutes.

Definition 2.1.12. If C and I are categories, with I small, then we denote by CI the category
whose objects are functors from I to C, and whose morphisms are natural transformations
between said functors. The objects of the functor category CI are often referred to in the
literature as I-indexed diagrams in C. Two diagrams D, D0 ∈ CI are said to be isomorphic,
denoted D ∼= D0, if they are naturally isomorphic as functors.

Here is an example describing the main type of indexing category we will consider

throughout the dissertation.
Definition 2.1.13. A partially ordered set, or poset, is a set P with a relation (cid:22) satisfying
the following properties:

1. Reflexivity: for each x ∈ P, we have x (cid:22) x.

2. Transitivity: for each x, y, z ∈ P satisfying x (cid:22) y and y (cid:22) z, it is also true that x (cid:22) z.

3. Antisymmetry: for each x, y ∈ P satisfying x (cid:22) y and y (cid:22) x, it is true that x = y.

Example 2.1.14. Let (P,(cid:22)) be a partially ordered set (i.e., a poset). Let P denote the
category whose objects are the elements of P, and a unique morphism x → y for each pair
x (cid:22) y in P. We call P the poset category of P.

18

This category is both small and thin and will be used for the purpose of indexing category
I. As for the target category C, we will mostly be interested in spaces and their algebraic
invariants.

2.1.3 Coequalizers

The contents of this section will be useful for the theory of homotopy colimits in section
2.2.2.

Definition 2.1.15. The undercategory of c ∈ C, denoted (c ↓ C), is the category whose
objects are pairs (c0, σ) consisting of an object c0 ∈ C and a morphism σ : c → c0 in C. A
morphism from (c0, σ : c → c0) to (c00, β : c → c00) is a morphism τ : c0 → c00 in C such that
τ ◦ σ = β.

Remark 2.1.16. If σ : c → c0 is a morphism in C, then composing with σ induces covariant
functors σ∗ : (c0 ↓ C) → (c ↓ C) and σ∗

op : (c0 ↓ C)op → (c ↓ C)op.

Definition 2.1.17. Let {Ai}i∈I be a family of objects in a category C. The product of this
family, if it exists, is an object P ∈ C along with morphisms pi : P → Ai such that for any
Y ∈ C and any collection of morphisms {yi : Y → Ai}i∈I, there exists a unique morphism
f : Y → P so that pi ◦ f = yi for each i. The product, since when it exists it is unique up
to isomorphism, will be denoted ×
Ai. The existence of f for any Y and any {yi}i∈I, is

referred to as the universal property defining the categorical product.

i∈A

The dual notion of coproduct of {Ai}i∈I is the object C ∈ C, when it exists, along
with morphisms ci : Ai → C such that for any object Y and any collection of morphisms
{yi : Ai → Y }i∈I, there exists a unique morphism g : C → Y such that g ◦ ci = yi for each i.
The coproduct of {Ai}i∈I is denoted ‘
Ai, and the existence of g is the universal property

i∈I

defining it.

19

Example 2.1.18. If Ai ∈ Top, then the product ×
is the Cartesian product endowed with the product topology, while the coproduct ‘

Ai is the space whose underlying set
Ai is

i∈A

their disjoint union.

i∈I

Definition 2.1.19. Let f, g : A → B be morphisms in a category C. The coequalizer of f
and g, denoted coeq(A→→B) if it exists, is an object C ∈ C with a morphism q : B → C such
that q ◦ f = q ◦ g. Moreover, if there is another pair (C0, q0) such that q0 ◦ f = q0 ◦ g, then
there is a unique u : C → C0 satisfying u ◦ q = q0.

Example 2.1.20. If f, g : A → B are two morphisms in Top, then coeq(A→→B) is the
quotient space B

.(cid:16)
f(a) ∼ g(a), ∀a ∈ A

(cid:17).

2.2 Algebraic Topology

In this section, we provide a brief review of the classical Künneth formula for the product
of topological spaces with coefficients in a Principal Ideal Domain (PID) in 2.2.1, an explicit
model for the homotopy colimit of a diagram of topological spaces in 2.2.2, and the theory
of principal bundles and Čech cohomology in 2.2.3.

2.2.1 The Classical Künneth Theorems

The classical Künneth theorem in algebraic topology relates the homology of the product
of two spaces to the homology of its factors. The relation is via a split natural short exact
sequence, and the proof is a combination of the algebraic Künneth formula for chain com-
plexes, and the Eilenberg-Zilber theorem. Both theorems are stated next, but first here are
two relevant definitions:
Definition 2.2.1. Let R be a commutative ring, and let C, C0 ∈ ChR. The tensor product
chain complex C ⊗R C0 consists of R-modules (C ⊗R C0)n =L
n−i) and boundary

(Ci ⊗R C0

i

20

morphisms

n−i −→ (C ⊗R C0)n

Ci ⊗R C0
c ⊗R c0

7→ ∂ic ⊗R c0 + (−1)ic ⊗R ∂0

n−ic0.

Definition 2.2.2. An R-module M is called flat, if for every short exact sequence 0 → A →
A0 → A00 → 0 of R-modules, the induced sequence

0 → A ⊗R M → A0 ⊗R M → A00 ⊗R M → 0

is also exact.

In particular, all free modules are flat.

Theorem 2.2.3 (The Algebraic Künneth Formula). If R is a PID and at least one of the
chain complexes C, C0 is flat (i.e., the constituent modules are flat), then for each n ∈ N0
there is a natural short exact sequence

0 −→ M

i+j=n

(Hi(C) ⊗R Hj(C0)) −→ Hn(C ⊗R C0) −→

M

TorR(Hi(C), Hj−1(C0)) −→ 0

which splits but not naturally.

i+j

See for instance [40, Chapter 5, Theorem 2.1]. It is well known that when C and C0 are
the singular chain complexes of two topological spaces X and Y , denoted by S∗(X; R) and
S∗(Y ; R) respectively, then the Eilenberg-Zilber theorem [32] provides a link between the
algebraic Künneth theorem and the homology of X × Y :

Theorem 2.2.4 (Eilenberg-Zilber). For topological spaces X and Y , there is a natural chain
equivalence ζ : S∗(X; R) ⊗R S∗(Y ; R) −→ S∗(X × Y ; R), and thus

Hn(X × Y ; R) ∼= Hn

for all n ≥ 0.

(cid:16)

S∗(X; R) ⊗R S∗(Y ; R)(cid:17)

21

The existence of ζ is in fact an application of the acyclic models theorem of Eilenberg and
MacLane [30]. This theorem provides conditions under which two functors to the category of
chain complexes produce naturally isomorphic homology theories. Later, we will use similar
ideas to prove a persistent Eilenberg-Zilber theorem (Theorem 3.3.10), which then yields our
persistent Künneth formula for the tensor product (Theorem 3.3.12). For now, here is the
topological Künneth theorem:

Corollary 2.2.5 (The topological Künneth formula). Let X, Y be topological spaces and let
R be a PID. Then, there exists a natural short exact sequence

(Hi(X; R) ⊗R Hj(Y ; R)) −→ Hn(X × Y ; R) −→

M

TorR(Hi(X; R), Hj−1(Y ; R)) −→ 0

(2.1)

0 −→ M

i+j=n

which splits, though not naturally.

i+j=n

For the sake of completeness, here is an example using the topological Künneth formula

to compute the homology of a product of topological spaces:

Example 2.2.6. Let X = S1 = Y and R = Z. Then the homology modules are Hi(X; Z) =
Z = Hi(Y ; Z) for i = 0, 1 and {0} otherwise. Since TorZ is {0} for any free input, we obtain
that

H0(X × Y ; Z) ∼= H0(X; Z) ⊗Z H0(Y ; Z) = Z ⊗Z Z ∼= Z
H1(X × Y ; Z) ∼= H0(X; Z) ⊗Z H1(Y ; Z) ⊕ H1(X; Z) ⊗Z H0(Y ; Z) ∼= Z ⊕ Z
H2(X × Y ; Z) ∼= H1(X; Z) ⊗Z H1(Y ; Z) = Z ⊗Z Z ∼= Z

which is indeed the homology of a 2-torus.

2.2.2 Homotopy Colimits

The machinery of homotopy colimits will be used in Section 3.1 to define the generalized
tensor product ⊗g : TopN × TopN −→ TopN appearing in Theorem 3.3.12. We provide

22

here a basic review, but direct the interested reader to [27] or [73] for a more comprehensive
presentation. If C is a category, then let BC denote its classifying space—i.e., the geometric
realization of the nerve of C.
Definition 2.2.7. For a small indexing category I, let D : I −→ Top be a functor. The
colimit of D is defined as:

colim(D) := coeq

(2.2)

a

i→j

D(i)→→a

i



D(i)

a

i→j

D(i) × B(j ↓ I)op→→a

i



where the top map sends D(i) to itself via the identity, the bottom map sends D(i) to D(j)
via D(i → j), and the first coproduct is indexed over all morphisms in I. Similarly, the
homotopy colimit of D is defined as:

hocolim(D) := coeq

D(i) × B(i ↓ I)op

(2.3)

The top map D(i) × B(j ↓ I)op → D(j) × B(j ↓ I)op is D(i → j) times the identity, while
the bottom map D(i)×B(j ↓ I)op → D(i)×B(i ↓ I)op is the identity of D(i) times the map
on classifying spaces induced by (i → j)∗

op : (j ↓ I)op → (i ↓ I)op.

It is worth noting that if D, D0 ∈ TopI, then any morphism D → D0 induces (functori-
ally) a map hocolim(D) → hocolim(D0), and that if f ∈ JI and D ∈ TopJ, then f induces
a natural map φf : hocolim(D ◦ f) → hocolim(D) called the change of indexing category.
Moreover, we have the following theorem from [27]:
Proposition 2.2.8. Let f : I → J, g : J → K and D : K → Top be functors. Then
φg◦f = φg ◦ φf.

While both the colimit and the homotopy colimit of a diagram of spaces yield methods
for functorially gluing spaces along maps, only the latter preservers homotopy equivalences.
Indeed, [73, Proposition 3.7] says:
Theorem 2.2.9. Let D, D0 ∈ TopI. If D → D0 is a natural (weak) homotopy equivalence—
i.e. each D(i) → D0(i) is a (weak) homotopy equivalence—then the induced map hocolim(D) →
hocolim(D0) is a (weak) homotopy equivalence.

23

When the indexing category has a terminal object, that is, a unique object z such that
for all i ∈ I there is a unique morphism i → z, then the homotopy colimit is particularly
simple [27, Lemma 6.8].

Theorem 2.2.10. Suppose that I has a terminal object z. Then for all D ∈ TopI, the
collapse map hocolim(D) → D(z) is a weak homotopy equivalence.

Notice that by collapsing the classifying spaces B(j ↓ I)op and B(i ↓ I)op in the definition
of homotopy colimit to a point, we recover the definition of the colimit. Next, we will see
specific conditions under which this collapse defines a homotopy equivalence from hocolim(D)
to colim(D). In order to state the result, we recall the notion of a cofibration.

Definition 2.2.11. A continuous map f : A → X is called a cofibration if it satisfies
the homotopy extension property with respect to all spaces Y . That is, given a homotopy
ht : A → Y and a map H0 : X → Y such that H0 ◦ f = h0, then there is a homotopy
Ht : X → Y such that Ht ◦ f = ht for all t. If f(A) is a closed subset of X, then f is called
a closed cofibration.

Remark 2.2.12. Let X ∈ TopN, and let T (X ) ∈ TopN be the functor defined as follows.
For j ∈ N0, let Tj(X ) be the mapping telescope of

X0

f1−→ ··· fj−2−−−→ Xj−1
f0−→ X1
Explicitly, Tj(X ) is the quotient space
  G
!(cid:30)

Tj(X ) = Xj × {j} t

fj−1−−−→ Xj.

Xi × [i, i + 1]

(x, i + 1) ∼ (fi(x), i + 1).

i<j

The functor T (X ) ∈ TopN satisfies that Ti(X ) ,→ Tj(X ) is a closed cofibration for all i ≤ j
[10, Chapter VII, Theorem 1.5].

Here is one situation where colim and hocolim provide similar answers,

24

Lemma 2.2.13 (Projection Lemma). Let P be a partially ordered set and P its poset cat-
egory. Let D ∈ TopP be such that D(i (cid:22) j) : D(i) → D(j) is a closed cofibration for each
pair i (cid:22) j in P. Then the collapse map

hocolim(D) → colim(D)

is a homotopy equivalence.

Proof. See [73, Proposition 3.1].

2.2.3 Principal bundles and Čech cohomology

We present here a review of the theory of principal bundles and their connection with Čech
cohomology. For details, see [43]. Throughout this section, we assume that B is a connected
and paracompact topological space with basepoint b0 ∈ B.
Definition 2.2.14. A pair (p, E) with E a topological space and p : E → B a continuous
map, is called a fiber bundle over B with fiber F = p−1(b0), if the following are satisfied:

1. p is surjective,

2. Every point b ∈ B has an open neighborhood U ⊂ B and a homeomorphism ρU :
U × F → p−1(U), called a local trivialization around b, such that p ◦ ρU(b0, e) = b0 for
every (b0, e) ∈ U × F.

The spaces E and B are called the total space and the base space of the bundle respectively,
and p is called the projection map.

Let (G, +) be an additive abelian topological group, that is, a topological space G with

a continuous additive group structure.

25

Definition 2.2.15. A (topological) right action of a topological group G on a topological
space X is a continuous map

· : X × G → X

(x, g) → x · g

satisfying

1. x · 1G = x for all x ∈ X, and
2. (x · g1) · g2 = x · (g1 + g2) for all x ∈ X and g1, g2 ∈ G.

The action is called transitive if for every x, y ∈ X, ∃g ∈ G, such that y = x · g. It is called
free if for all g 6= h ∈ G, ∃x ∈ X, such that x · g 6= x · h.

Definition 2.2.16. A principal G-bundle is a fiber bundle (p, E) so that E is equipped with
a right G-action satisfying

1. · : E × G → E is fiberwise preserving, that is,

p(e · g) = p(e) ∀(e, g) ∈ E × G.

2. The induced fiberwise G-action p−1(b) × G → p−1(b) is free and transitive for every b

in the base space B.

3. The local trivializations ρU : U × F → p−1(U) can be chosen to be G-equivariant, that

is, they satisfy

ρU(b, e · g) = ρU(b, e) · g

for every (b, e, g) ∈ U × F × G.

Definition 2.2.17. Two principal G-bundles pj : Ej → B (for j = 1, 2) are called isomor-
phic, if there exists a G-equivariant homeomorphism Φ : E1 → E2 such that p2 ◦ Φ = p1.

26

This defines an equivalence relation on the set of principal G-bundles over B, and the set

of its equivalence classes (isomorphism classes) is denoted by PrinG(B).
Remark 2.2.18. Given a principal G-bundle p : E → B and a system of G-equivariant local
trivializations

{ρj : Uj × F → p−1(Uj)}j∈J ,

we have that ρ−1
k ◦ ρj : (Uj ∩ Uk) × F → (Uj ∩ Uk) × F is a G-equivariant homeomorphism
whenever Uj ∩ Uk 6= ∅. Since the G-action on E is free and transitive on each fiber, the
composition ρ−1
k ◦ ρj induces a well defined continuous map ρjk : Uj ∩ Uk → G for each
j, k ∈ J, defined as:

ρ−1
k ◦ ρj(b, e) = (b, e · ρjk(b))

(2.4)
for all (b, e) ∈ (Uj ∩ Uk)× F. The maps ρjk are called the transition functions corresponding
the system of local trivializations {ρj}j∈J.

Now that we have the basics of principal bundles in place, we review some sheaf theory:

Definition 2.2.19. A presheaf F of groups over a topological space X consists of the
following data:

1. A group F(U) for each open set U ⊂ X, such that F(∅) = {0}, and

each V ⊂ U such that γU

W ◦ γU
V .

2. A collection of group homomorphisms, called restriction maps, γU
U = 1F(U) and if W ⊂ V ⊂ U, then γU

V : F(U) → F(V ) for
W = γV
If further, F satisfies the following gluing axiom, then it is called a sheaf :
If U ⊂ X is open, {Uj}j∈J is an open cover of U, and there exists a collection {sj,F(Uj)}j∈J
where sj ∈ F(Uj), satisfying

for all j, k ∈ J, then ∃ s ∈ F(U) such that

Uj
Uj∩Uk
γ

(sj) = γ

Uk
Uj∩Uk

(sk)

γU
Uj

(s) = sj

27

for each j ∈ J.

We now present a relevant example of a sheaf:

Example 2.2.20. If G is a topological group, and Maps(U, G) denotes the set of continuous
maps from U to G, then the sheaf CG of continuous G-valued functions over a topological
space X is defined as:

CG(U) := Maps(U, G).

The restriction maps CG(U) → CG(V ) are induced by inclusions V ,→ U. This satisfies
the gluing axiom because for an open cover {Uj}j∈J of an open set U ⊂ X, a family of
continuous functions {Uj → G}j∈J that agree on intersection of their domains can be glued
together to define a global continuous function U → G.

Definition 2.2.21 (Čech cohomology). Let F be a presheaf of abelian groups on a topolog-
ical space X and U = {Uj}j∈J be an open cover of X. Define the n-th Čech cochain groups
as

ˇCn(U;F) =

and the n-th coboundary map as

F(Uj0···jn)

(j0,j1,...,jn)∈Jn+1

Y

dn : ˇCn(U;F) → ˇCn+1(U;F)

where

dn(f)j0···jn+1 =

n+1X

k=0

(−1)kfj0···ˆjk···jn+1

|Uj0···jn+1

.

It can be checked that dn ◦ dn−1 = 0 for each n ≥ 1, that is each dn is indeed a coboundary
map. Define two subgroups of ˇCn(U;F), namely

ˇZn(U;F) := Ker(dn)
ˇBn(U;F) := Img(dn−1)

28

called the set of Čech n-cocycles and Čech n-coboundaries respectively. Their quotient is
the Čech n-th cohomology of the covering U:

ˇHn(U;F) := ˇZn(U;F)/ ˇBn(U;F).

The Čech cohomology of the topological space X is defined using direct limits of refine-

ments of open covers, which we define next:

Definition 2.2.22. Let U = {Uj}j∈J and V = {Vr}r∈R be open covers of X. We say that
V is a refinement of U, written as V ≤ U if ∃τ : R → J such that Vr ⊂ Uτ(r) for all r ∈ R.

Remark 2.2.23. The refinement function τ induces a chain map between Čech complexes

τ# : ˇCn(U;F) → ˇCn(V;F)

and hence a map between the cohomology groups

τ∗ : ˇHn(U;F) → ˇHn(V;F).

Furthermore, any other refinement function ˜τ : R → J induces the same homomorphism
between homology groups as τ. The Čech cohomology of a topological space X is defined as
the direct limit

over all covers with respect to refinement. In other words, this cohomology group is equal

ˇHn(X;F) := lim−→U

ˇHn(U;F)

to the quotient ‘

U

ˇHn(U;F)/ ∼ where

ˇHn(U;F) 3 [f] ∼ [g] ∈ ˇHn(V;F)

if and only if ∃ a common refinement W ≤ U,V such that τ∗U[f] = τ∗V[g].

We now transition (wordplay intended) back to principal bundles. The transition func-
tions {ρjk} defined in equation 2.4 also define an element ρ = {ρjk} ∈ ˇC1(U; CG) in the Čech
1-cochains of the cover U = {Uj}j∈J with coefficients in the sheaf CG defined in example
2.2.20. Furthermore, the element ρ is a Čech cocycle:

29

[

j∈J

Eη =

Uj × {j} × G

.(b, j, g) ∼ (b, k, ηjk(b))

Proposition 2.2.24. The transition functions ρjk satisfy the following cocycle condition:

ρjl(b) = (ρjk + ρkl)(b)

for all b ∈ Uj ∩ Uk ∩ Ul. In other words, {ρjk} ∈ ˇZ1(U; CG) is a Čech cocycle.

For any open cover U = {Uj}j∈J of a paracompact B and a Čech cocycle η := {ηjk} ∈

ˇZ1(U; CG), we can construct a principal G-bundle p : Eη → B as

for b ∈ Uj ∩ Uk. The projection p takes the class [(b, j, g)] to b. Furthermore,

Theorem 2.2.25. The function

is a natural bijection.

Proof. See 2.4 and 2.5 in [55].

ˇH1(B, CG) → PrinG(B)

[η] → [Eη]

For each topological group G, we have a weakly contractible space EG along with a group

action of G on it. The quotient space defined as

BG := EG/G

is called the classifying space of G, and the quotient map p : EG → BG defines a principal
G bundle called the universal bundle. For example, if G = Z, then EZ ’ R and BZ ’ S1,
and if G = Zq, then EZq ’ C∞ and BZq ’ L∞
q . There are many constructions of the space
EG, for example by Milnor [50] or Segal [63], but they are homotopy equivalent.

Furthermore, if we let h : B → BG be a continuous map, then

h∗EG := {(b, e) ∈ B × EG | h(b) = p(e)} → B
(b, e) → b

is another principal G bundle, called the pullback bundle. Then we have another bijection:

30

Theorem 2.2.26. The function

[B, BG] → PrinG(B)
[h] → [h∗EG]

is a natural bijection.

Proof. See [43, Chapter 4: Theorems 12.2 and 12.4].

Putting things together, we obtain that there is a natural bijection between ˇH1(B, CG)
and [B, BG]. Furthermore, if G is abelian and B is a CW complex, then the 1-Čech coho-
mology group ˇH1(B, CG) is isomorphic to the singular cohomology group H1(B; G). Before
we finish this section, we quickly want to describe Eilenberg-MacLane spaces:

Definition 2.2.27. A connected CW complex K(G, n) is called an Eilenberg-MacLane space
if its homotopy type is uniquely determined by its jth homotopy groups:



πj(K(G, n)) ∼=

0

if

j 6= n

G if

j = n.

For example, K(Z, 1) ’ S1, K(Z2, 1) ’ RP∞, K(Zq, 1) ’ L∞

q , and K(Z, 2) ’ CP∞.
If n = 1, then for an abelian discrete G, we have BG ’ K(G, 1). This has the following
consequence: Given a CW complex B, the Brown representability theorem for CW complexes
and singular cohomology (in dimension 1) gives a natural bijection H1(B; G) ∼= [B, K(G, 1)]
between the 1-st cohomology of B with coefficients in G and homotopy classes of continuous
maps from B to K(G, 1). This bijection was exploited for G = Z to construct the sparse
circular coordinates [55] and we study their stability in Chapter 5.

2.3 Persistent Homology

Persistent homology is an algebraic and computational tool from topological data analysis
[56]. Broadly speaking, it is used to quantify multiscale features of shapes. The successes

31

of persistent homology stem in part from a strong theoretical foundation [76, 24, 52] and a
focus on efficient algorithmic implementations [5, 6, 9, 20, 72].

The framework we use here is that of diagrams in a category. Such a framework for
persistence was first introduced by Bubenik and Scott in [13]. Perhaps the most important
takeaways from this section are the classification via barcodes, and that the persistent ho-
mology of a diagram of spaces is isomorphic to the standard homology of the associated
persistence chain complex (see [76] and Theorem 2.3.21).

The following is an example of a functor category defined in Definition 2.1.12:

Example 2.3.1. If Top is the category of topological spaces and continuous maps, and I
is a thin category, then each object of TopI is a collection X = {fj,i : Xi → Xj}i→j∈I of
topological spaces Xi, and continuous maps fj,i : Xi −→ Xj for each morphism i → j in
I satisfying: fi,i is the identity of Xi, and fk,j ◦ fj,i = fk,i for any i → j → k. Similarly,
a morphism φ from X = {fj,i : Xi → Xj} to Y = {gj,i : Yi → Yj} is a family of maps
φi : Xi → Yi such that gj,i ◦ φi = φj ◦ fj,i for every i → j.
Example 2.3.2. Let N0 = {0, 1, 2, . . .} with its usual (total) order, and let N be its poset
category. N-indexed diagrams in Top arise in Topological Data Analysis (TDA) as follows.
Let M be a metric space, let X ⊂ M be a subspace and let X ⊂ M be finite. In applications X
is the data one observes (e.g., images, text documents, molecular compounds, etc), obtained
by sampling from/around X (the ground truth, which is unknown in practice), both sitting
in an ambient space M. With the goal of estimating the topology of X from X, one starts
by letting X(
) be the union of open balls in M of radius 
 ≥ 0 centered at points of X.
Hence X(
) provides a—perhaps rough—approximation to the topology of X for each 
 ≥ 0,
and any discretization 0 = 
0 < 
1 < ··· of [0,∞) yields an object in TopN as follows:
(
j) is the map associated to i ≤ j. Such
N0 3 i 7→ X(
i), and the inclusion X(
i) ,→ X
objects allow one to avoid optimizing the choice of a single 
, leading to several recovery
theorems and algorithms for topological inference [52].

32

Remark 2.3.3. If each fi : Xi → Xi+1 in X ∈ TopN is an inclusion, e.g. as in Example 2.3.2,
then X defines a filtered topological space. Recall that a filtered topological space consists of
a space X together with a filtration X0 ⊂ X1 ⊂ ··· ⊂ X.

This remark motivates the following definition.

Definition 2.3.4. We call X ∈ TopN a filtered space if all its maps are inclusions. The
collection of all filtered spaces forms a full subcategory (see definition 2.1.8) of TopN denoted
FTop.

Example 2.3.5. Since the functor T (X ) ∈ TopN, defined in Remark 2.2.12, satisfies
Tj(X ) ⊂ Tk(X ) whenever j ≤ k, it defines an object in FTop called the telescope filtra-
tion of X .

As is typical in algebraic topology, one is interested in the interplay between diagrams of

spaces and diagrams of algebraic objects.

Example 2.3.6. Let R be a commutative ring with unity, and let ModR be the category
of (left) R-modules and R-morphisms as in Example 2.1.3. The typical objects in ModI
R
one encounters in TDA arise from objects X ∈ TopI by fixing n ∈ N0 and taking singular
homology in dimension n with coefficients in R. Indeed, Hn(Xi; R) is an R-module for each
i ∈ I, and for each morphism i → j in I, the map X (i → j) : Xi −→ Xj induces—in a
functorial manner—a well-defined R-morphism from Hn(Xi; R) to Hn(Xj; R). The resulting
object in ModI

R will be denoted Hn(X ; R).

Definition 2.3.7. Let M,N ∈ ModI
sending i ∈ I to (M ⊕ N )(i) := Mi ⊕ Ni, and each morphism i → j in I to

R, and let M ⊕ N : I −→ ModR be the functor

(M ⊕ N )(i → j) := M(i → j) ⊕ N (i → j).
R is indecomposable if M ∼= N ⊕ O only when either N or O is the
We say that M ∈ ModI
zero functor. Similarly, let M ⊗ N : I −→ ModR be the functor sending i ∈ I to Mi ⊗R Ni

33

and i → j to M(i → j) ⊗R N (i → j), where the tensor product ⊗R is the usual one for
R-modules and R-morphisms.

Example 2.3.8. Recall that an interval in a poset P is a set I ⊂ P for which i, k ∈ I and
i (cid:22) j (cid:22) k always imply j ∈ I. An interval I ⊂ P defines an object 1I in ModP
R as follows:
for j (cid:22) j0 in P, let

 R if j ∈ I

0 else

1I(j) :=

and

1I(j (cid:22) j0) :=

 idR if j, j0 ∈ I

else

0

The 1I’s are called interval diagrams, and if P is totally ordered, then they are indecompos-
R [13, Lemma 4.2]. Moreover, if I, J ⊂ P are intervals, then the tensor product
able in ModP
of the corresponding interval diagrams satisfies

1I ⊗ 1J

∼= 1I∩J .

Indeed, 1I ⊗ 1J(r) ∼= 0 if r /∈ I ∩ J, and 1I ⊗ 1J(r) ∼= R ⊗R R ∼= R if r ∈ I ∩ J, where
the last isomorphism is given by multiplication in R. Since applying idR ⊗R idR and then
multiplying equals multiplying and then applying idR, the result follows.
Definition 2.3.9. An object V ∈ ModIF satisfying the condition dimF V(j) < ∞ for all
j ∈ I is said to be pointwise finite dimensional.

Interval diagrams are fundamental building blocks in ModPF [22, Theorem 1.1]:

Theorem 2.3.10. Let F be a field and let (P,(cid:22)) be a totally ordered set. Suppose that P is
separable with respect to the order topology, that is it contains a countable dense subset, and
that V ∈ ModPF is pointwise finite dimensional. Then, there exists a multiset of intervals
I ⊂ P called the barcode of V, denoted bcd(V), and so that

V ∼= M

1I .

I∈bcd(V)

Moreover, bcd(V) only depends on—and uniquely determines—the isomorphism type of V.

34

Definition 2.3.11. Let F and P be as in Theorem 2.3.10, and let X ∈ TopP be so that
Hn(X ; F) is pointwise finite. The barcode of Hn(X ; F), denoted bcdn(X ; F), is the unique
multiset of intervals in P such that

Hn(X ; F) ∼= M

1I .

I∈bcdn(X ;F)

It follows that bcdn(X ; F) provides a succinct description of the isomorphism type of
Hn(X ; F). The design of algorithms for the computation of barcodes, at least in the N0-
indexed case, leverages a more concrete description of Hn(X ; F) which we describe next.
Definition 2.3.12. For M = {hj,i : Mi −→ Mj}i(cid:22)j∈P an object in ModP

R, let

PM := M

i∈P

Mi.

(2.5)

PM is called the persistence module associated to M.

The word module stems from the following observation: if M ∈ ModN

R , then PM is a
graded module over R[t], the graded ring of polynomials in a variable t. Indeed, the R[t]-
module structure is defined through multiplication by t as follows: for m = (m0, m1, . . .) ∈
PM let

t · m :=(cid:16)0, h0(m0), h1(m1), . . .

(cid:17)

and extend the action to R[t] in the usual way. The graded nature of the multiplication
comes from noticing that tk · Mi ⊂ Mi+k for every i, k ∈ N0.

If φ : M −→ N is a morphism in ModN

R , then it can be readily checked that P φ :=

⊕iφi : PM −→ PN is a graded R[t]-morphism, and that P defines a functor satisfying:

Theorem 2.3.13 (Correspondence [76]). Let gModR[t] denote the category of graded R[t]-
R −→ gModR[t] is an equivalence of
modules and graded R[t]-morphisms. Then, P : ModN
categories.

A generalization of this result appeared in [21] which drops the condition on commuta-
tivity of R and asserts that the correspondence is an isomorphism instead of an equivalence.

35

Remark 2.3.14. Let ‘ < ρ in N0 ∪ {∞}, and let 1[‘,ρ) ∈ ModN
diagram. Then,

P 1[‘,ρ) ∼=(cid:16)

t‘ · R[t](cid:17)

/ (tρ)

R be the resulting interval

as graded R[t]-modules, with the convention t∞ = 0. These are called interval modules.

Let X ∈ TopN—for instance, encoding multiscale approximations to the topology of an
underlying unknown space (see Example 2.3.2). Applying Hn( · ; R) as in Example 2.3.6
yields an object in ModN
R , and applying the functor P from the Correspondence Theorem
(2.3.13) defines a graded R[t]-module. Explicitly:

Definition 2.3.15. Given an object X ∈ TopN, its n-dimensional persistent homology with
coefficients in R is the persistence module P Hn(X ; R) ∈ gModR[t],

P Hn(X ; R) := M

Hn(Xi; R).

Remark 2.3.16. If F is a field and Hn(X ; F) is pointwise finite, then

i∈N0
M

(cid:16)
t‘ · F[t](cid:17)

/ (tρ)

P Hn(X ; F) ∼=

[‘,ρ)∈bcdn(X ;F)

as graded F[t]-modules. A graded version of the structure theorem for modules over a PID
implies that if P Hn(X ; F) is finitely generated over F[t], then bcdn(X ; F) can be recovered
from the (graded) invariant factor decomposition of P Hn(X ; F). This is how the first general
persistent homology algorithms were implemented [76].

Example 2.3.17. Recall that a simplicial complex is a collection K of finite nonempty
sets—called simplices—so that if σ ∈ K and ∅ 6= τ ⊂ σ, then τ ∈ K. Let K(n) ⊂ K be the
collection of simplices of K with n + 1 elements—these are called n-simplices and we write
v ∈ K(0) instead of {v} ∈ K(0). A function f : K −→ K0 between simplicial complexes is
called a simplicial map if f({v0, . . . , vn}) = {f(v0), . . . , f(vn)} for every {v0, . . . , vn} ∈ K.

Because simplicial complexes carry more structure than arbitrary topological spaces, they
are easier to code in computers. As a result in TDA, instead of diagrams in Top, diagrams

36

in Simp are used. Using isomorphisms of singular and simplicial homology, one can show
that the entire persistent homology algorithm works for the case of simplicial complexes
(as we will see in remark 3.2.1). The following example illustrates the theory of persistent
homology:

Example 2.3.18. Consider the filtration of a simplicial circle defined in figure 2.1. The

Figure 2.1: A filtered simplicial circle K = {Ki}i, with Ki = K5 for i ≥ 5.

barcodes are bcd(K) = {[0,∞)0, [1, 3)0, [2, 4)0, [5,∞)1}, where the subscripts denote homo-
logical dimension.

C∗ ⊕ C0∗ =n

Thus far we have described persistent homology in terms of barcodes and F[t]-graded
modules. The next, and final description, is in terms of the standard homology of the persis-
tence chain complex. Indeed, let ChR denote the category of (N0-graded) chain complexes
of R-modules, and chain maps. Recall that given two chain complexes C∗ and C0∗, their
direct sum is given by

o
i∈N0
Definition 2.3.19. Let C∗ = {fj : C∗j −→ C∗j+1} be an object in ChN
R . That is, each C∗j
is a chain complex of R-modules, and the fj’s are chain maps. Then, the persistence chain
complex of C∗ is
where each PCi = L

Ci,j ∈ gModR, and therefore PC∗ is an object in the category

∂i ⊕ ∂0

i : Ci ⊕ C0

i −→ Ci−1 ⊕ C0
i−1

.

PC∗ := M

C∗j

j∈N0

gChR[t] of chain complexes of graded R[t]-modules.

j∈N0

37

Remark 2.3.20. Since homology commutes with direct sums of chain complexes, then

Hn(C∗j) = P Hn(C∗)

Hn(PC∗) ∼= M

j∈N0

as R[t] modules.

For X ∈ Top, let S∗(X; R) ∈ ChR denote the chain complex of singular chains in X
with coefficients in R. Then, given X ∈ TopN, composition of functors yields an object
S∗(X ; R) in ChN
R . The associated persistence chain complex is thus an object P S∗(X ; R)
in gModR[t] and its homology—by Remark 2.3.20—recovers the persistent homology of X .
In other words,

Theorem 2.3.21 ([76]). Let X ∈ TopN. Then, its persistent homology is isomorphic over
R[t] to the homology of P S∗(X , R):

P Hn(X ; R) ∼= Hn(P S∗(X ; R)).

(2.6)

For a general (small) indexing category I the barcode is no longer available as a discrete
invariant for pointwise finite objects in ModIF [17]. However, it is always possible to consider
the rank invariant:

Definition 2.3.22. Let M ∈ ModIF . The rank invariant of M is the function on morphisms
of I given by ρM(i → j) = rank(M(i → j)) ∈ N0 ∪ {∞}. For an object X ∈ TopI we let
n := ρHn(X ;F).
ρX
Remark 2.3.23. Note that ρM is an invariant of the isomorphism type of M. It is in fact
computable in polynomial time when I = Nk and PM is finitely generated as an F[t1, . . . , tk]
module [16]. If M ∈ ModRF is pointwise finite where R is the poset category of R, then
ρM and bcd(M) can be recovered from each other [17, Theorem 12]. Furthermore, ρM is a
complete invariant for the isomorphism type of M if I is a subcategory of R.

38

2.4 Dynamical Systems

In this section, we cover the necessary basics of dynamical system needed for this disser-
tation. Simply put, a dynamical system consists of two components: a state space M and
an evolution rule φ that describes how these states change as a function of time. Perhaps,
one of the most popular examples in the study of dynamical systems is the Lorenz attractor.
It is a system of the following ordinary differential equations,

= σ(y − x)
= x(ρ − z) − y
= xy − βz

dx
dt
dy
dt
dz
dt

where x, y, and z are functions of time t and σ, β, and ρ are positive constants. Here, the
state space M is the space of all 3-tuples, that is R3, and the evolution φ is dictated by the
solution to the ODE system. It was originally developed by Edward Lorenz as a simplified

Figure 2.2: The Lorenz attractor for parameters σ = 10, β = 8

3, and ρ = 28.

mathematical model for atmospheric convention [47], and became widely known for having
chaotic solutions for certain parameters and initial conditions.

39

Next, we give a definition of dynamical system on topological spaces:

Definition 2.4.1. A continuous time dynamical system is a topological space M along with
an evolution rule consisting of a continuous map φ : M × R → M satisfying the condition

φ(φ(x, t), s) = φ(x, t + s)

for all x ∈ M and t, s ∈ R.

Note that in the above definition, the topological space M serves as the space of states
mentioned at the beginning of this section. Next, we give another example of a dynamical
system:

Example 2.4.2. Recall that the 2-torus T2 can be defined as the quotient R2/Z2. An
important example of a dynamical system on T2 is that of the irrational flow:

φ : T2 × R → T2
((x, y), t) → (x + t, y + ωt)

where ω is irrational.

Figure 2.3: The evolution of (0, 0) under the irrational flow φ on T2 as defined in Example
2.4.2 for ω =

√
3.

40

If M is a differentiable manifold with a vector field X on it, then the flow associated to
X gives us an evolution rule. Restriction to a discrete subset of R, say N, yields a discrete
time dynamical system. Another example is of a manifold M and a diffeomorphism ϕ. In
this case,

φ : M × N → M

(x, n) → φ(x, n) := ϕn(x).

In practice, dynamical systems are present everywhere and often, the evolution of states
is captured via temporal data, i.e. a time series. To be more precise, given the evolution
function φ of a dynamical system, a real (or complex) valued function f : M → R called an
observation function, and a state x, a time series can be defined

F : R → R

t → f (φ(x, t)) .

For example, for the irrational flow in Example 2.4.2, define an observation function

f : T2 → R
(x, y) → cos(x) + cos(y).

Then for the observation of the evolution of (0, 0) under φ gives us a time series F(t) =
cos(t) + cos(ωt). See Figure 2.4.

So far, we have discussed that given a dynamical system, one can obtain relevant mea-
surements in the form of a time series. The next section discusses the conditions under
which given a time series, some information about the underlying dynamical system can be
recovered.

2.4.1 Takens’s embedding theorem

In 1981, Floris Takens proved a delay embedding theorem [67, Theorem 1] which provides
conditions under which a smooth attractor can be reconstructed from a sequence of obser-

41

Figure 2.4: Time series associated to the observation f of the evolution of (0, 0) for the
irrational flow φ.

vations made with a generic observation function. While this reconstructed attractor might
have a different geometric shape than the original, its topology is still preserved. We present
next the version of Takens’ theorem which is the most relevant to us, in particular, when
the manifold M is compact and the dynamical system is discrete in nature.

Theorem 2.4.3 ([67], Theorem 1). Let M be a m-dimensional compact Riemannian mani-
fold. For pairs (ϕ, f), where ϕ ∈ C2(M, M) and f ∈ C2(M, R), it is a generic property that
the map Φϕ,f : M → R2m+1 defined as

Φϕ,f(x) =(cid:16)

f(x), f(ϕ(x)), f(ϕ2(x)), . . . , f(ϕ2m(x))(cid:17)

is an embedding.

There are two things to note here. The first is that the genericity of the pair (ϕ, f) is
in C2-Whitney topology on the function space C2 (M, M × R), in which roughly speaking,
two functions f and g are close to each other if and only if their derivatives are close to each
other. For a more rigorous version, we refer the reader to [41]. The other is that Takens
also provides similar theorems for continuous dynamical systems and in the case when the

42

manifold M is non-compact. Takens’ theorem motivates the definition of the sliding window
embedding which is what we describe next.

Definition 2.4.4. For a function f : R → C, an integer d > 0 called the embedding
dimension and a real number τ > 0 called the time delay, the sliding window embedding of
f at t is given by:





SWd,τ f(t) =

f(t)

f(t + τ)

...

f(t + dτ)

If this function is indeed a time series generated by observing some underlying dynamical
system, Takens’ theorem implies that by studying the shape of the sliding window point
cloud, we can recover some information about the dynamical system. Let us look at some
examples, with a simple function g(t) = cos(t) first.

Example 2.4.5. Assume a particle at (1, 0) on a unit circle S1 is moving counter-clockwise
with the function φ(t) = (cos(t), sin(t)). Let p : S1 → R defined as p((x, y)) = x be the
observation. Then we get a time series

g(t) = p(φ(t)) = cos(t).

The sliding window embedding SWd,τ g(R) of g with appropriate dimension and delay re-
covers the topology of S1.

While the figure on the right is topologically a circle, it looks different from a standard
circle geometrically. The reader might ponder why it would be a problem—because in real
applications, we only have a discrete time series and the hope is that the sliding window
embedding of a continuous approximation to this time series recovers the topology of the un-
derlying dynamical system. In [58], Perea and Harer used Rips persistent homology to study
finite sliding window point clouds generated from general periodic functions and provided

43

Figure 2.5: Dynamical System to Sliding Window Reconstruction for a counterclockwise flow
on S1: In the leftmost figure, we see a standard circle. In the middle, we see the time series
generated g(t). In the rightmost figure, we see the reconstructed circle from sliding window
embedding.

conditions on parameters d and τ under which the geometry of the recovered topological
circle is a good geometric approximation. In particular, we will use persistent homology of
Rips complexes on SWd,τ g(T) ⊂ SWd,τ g(R) ⊂ Cd+1 for some T ⊂ R.

Next, we have another example of sliding window embedding of a periodic function

g(t) = cos(t) + cos(3t).

Example 2.4.6. Assume a particle at (0, 0) on T2 is moving with the function φ(t) = (t, 3t).
Let p : T2 → R defined as p((x, y)) = cos(x) + cos(y) be the observation. Then we get a
time series

g(t) = p(φ(t)) = cos(t) + cos(3t).

The sliding window embedding SWd,τ g(R) of g with appropriate dimension and delay re-
covers the topology of S1. Rips persistent homology confirms that in Figure 2.6.

Figure 2.6: Dynamical System to Persistence for a rational flow on T2: In the leftmost
figure, we see a rational flow on T2. In the second, we see the time series generated g(t). In
the third figure, we see a PCA visualization of the reconstructed circle from sliding window
embedding. In the rightmost figure, we see the 1- dimensional Rips persistence diagram of
a sliding window point cloud sampled from the third figure.

44

√

A closely related function to the periodic function in the example above is g(t) =
3t), which is an example of a quasiperiodic function, which we will study
cos(t) + cos(
in its generality in Chapter 4. In simpler terms, a quasiperiodic function is controlled by
two or more harmonics and these harmonics are linearly independent over Q. A special first
case was studied by Perea in [53].

√

Example 2.4.7. Assume a particle at (0, 0) on T2 is moving with the irrational flow function
3t). Let p : T2 → R defined as p((x, y)) = cos(x) + cos(y) be the observation.
φ(t) = (t,
Then we get a time series

g(t) = p(φ(t)) = cos(t) + cos(

√
3t).

The sliding window embedding SWd,τ g(R) of g with appropriate dimension and delay re-
covers the topology of T2. Rips persistent homology confirms that in Figure 2.7.

Figure 2.7: Dynamical System to Persistence for an irrational flow on T2: In the leftmost
figure, we see an irrational flow on T2. In the second, we see the time series generated g(t).
In the third figure, we see a PCA visualization of the reconstructed torus from sliding window
embedding. In the rightmost figure, we see the Rips persistence diagrams in dimensions 1, 2
of a sliding window point cloud sampled from the third figure.

Before finishing this section, we want to mention a property that we will use later. The

following was shown in [59].

Proposition 2.4.8. For all d ∈ N and τ > 0, the mapping SWd,τ : C(R, C) → C(R, Cd+1)
is a bounded linear operator with norm

|| SWd,τ ||≤ √

d + 1.

45

2.5 Analytical tools

In this section, we provide a brief review of Fourier series in 2.5.1, Kronecker’s approxi-

mation theorem in 2.5.2, and Wasserstein metric on probability measures in 2.5.3.

2.5.1 Fourier Series

Fourier series were introduced by Joseph Fourier in the 1800s for the purpose of solving
the heat equation in a metal plate. Using the ideas that Fourier introduced, it was shown
that every arbitrary real valued continuous periodic function can be approximated by a
sum of sines and cosines. Since then Fourier series have had many applications in electrical
engineering, acoustics, optics, signal & image processing, etc. Let us review:

Let T be the quotient space R/(2πZ). For a single variable periodic function g ∈ L2 (T)
and Z ∈ N0 = N∪{0}, its Fourier series and its Z-truncated Fourier polynomial are written
as

∞X

k=−∞

g(t) =

respectively, where

ZX

k=−Z

ˆg(k)eikt

ˆg(k)eikt

and SZ g(t) =

ˆg(k) = 1
2π

πZ

−π

g(t)e−iktdt

are the Fourier coefficients of g and the series given by partial sums SZ g converges to g
almost everywhere as Z → ∞. If additionally, g is continuously differentiable, then pointwise
convergence is guaranteed by the following pipeline of results in analysis:

Proposition 2.5.1.

1. If f ∈ C1([a, b]), then f, f0 are both continuous on [a, b] by defi-

nition. Then f is Lipschitz.

2. If f is Lipschitz on [a, b], then f is uniformly continuous and has bounded variation,

with the variation bounded by Lf(b − a) where Lf is the Lipschitz constant.

46

3. (Jordan’s Criterion) If f is a function of bounded variation in a neighbourhood of x,

then

Z→∞ SZ f(x) = 1
lim

2[f(x+) + f(x−)].

From this proposition, we can conclude that if g ∈ C1([a, b]), then SZ g(x) converges to

g(x) for all x ∈ [a, b]. A proof of these propositions can be found in [29, Chapter 10].

2.5.2 Kronecker’s theorems

Recall that R can be viewed as an infinite dimensional vector space over Q with an uncount-
able basis. A finite subset {β1, . . . , βN} ⊂ R of reals (vectors) is called incommensurate if
β1, . . . , βN are linearly independent over Q, that is,

r1β1 + ··· + rN βN = 0 ⇒ r1 = ··· = rN = 0

for r1, . . . , rN ∈ Q. The following theorem by Kronecker is a result about diophantine
approximations applying to several real numbers. A proof can be found in [42, Chapter 2].

Theorem 2.5.2 (Kronecker’s theorem). Let {β1, . . . , βN} be a set of real numbers, r1, . . . , rN
be arbitrary real numbers, and 
, L be arbitrary positive numbers, then one can find integers
l and p1, . . . , pN with

|βnl − rn − pn| < 


for

1 ≤ n ≤ N

where |l| ≥ L if and only if {β1, . . . , βN} is incommensurate. As a consequence, {β1, . . . , βN}
is incommensurate over Q if and only if the collection {(zβ1, . . . , zβN) | z ∈ Z} of N-tuples is
dense in TN = (R/(2πZ))N, which is equivalent to the collection {(eiβ1z, . . . , eiβN z)|z ∈ Z}
being dense in TN = S1 × ··· × S1.

In section 4.3, we will use Kronecker’s theorem to prove:

Corollary 4.3.1: An α-tuple β = (β1, . . . , βα) of real numbers spans a N-dimensional vector
space over Q if and only if the collection {zβ | z ∈ Z} is dense in an embedding of TN in

47

Tα.
which in turn will be essential to proving the structure theorem:
Theorem 4.3.5: The sliding window embedding SWd,τ SZ f(Z) is dense in a space homeo-
morphic to TN in Td+1 for appropriate conditions on τ, d, and Z.

2.5.3 Wasserstein metric on probability measures

In this section, we review the definition and relevant properties of the Wasserstein metric.
We will use this metric to measure the noise on the landmark sets in Chapter 5, for which
we prove the stability theorems.

To begin, consider a compact metric space (M, d).

Definition 2.5.3. A correspondence (or relation) between non-empty subsets A, B ⊂ M is
a subset R ⊂ A × B such that

• for all a ∈ A, there exists b ∈ B such that (a, b) ∈ R, and

• for all b ∈ B, there exists a ∈ A such that (a, b) ∈ R.

Denote by R(A, B) the set of all possible correspondences between A and B. Furthermore,
define the Hausdorff distance as

dH(A, B) =

inf

R∈R(A,B)

sup
(a,b)∈R

d(a, b).

Definition 2.5.4. Given A, B ⊂ M and measures µA, µB on M with Supp(µA) = A and
Supp(µB) = B, a measure µ on A × B is a coupling of µA and µB if and only if

µ(A0 × B) = µA(A0) ∀A0 ⊂ A measureable and
µ(A × B0) = µB(B0) ∀B0 ⊂ B measureable.

Denote by M(µA, µB), the collection of all couplings of µA and µB. Denote by R(µ) =
B =n
Supp(µ), the correspondence defined by the support of a coupling. For A = {ai} and

o finite, we represent a coupling as a |A| × |B| matrix [µij] with µij = µ(ai, bj).

bj

48

Definition 2.5.5. For p ≥ 1 or p = ∞, the following
Z

dW,p(µA, µB)p =

d(a, b)pdµ(a, b) and,

inf

µ∈M(µA,µB)

A×B

dW,∞(µA, µB) =

inf

µ∈M(µA,µB)

sup

(a,b)∈R(µ)

d(a, b)

define the Wasserstein–Kantorovich–Rubinstein metrics between measures on M.

Since the support of a coupling yields a relation, we have

Lemma 2.5.6. Let A, B ⊂ (M, d) compact metric space and let µA and µB be measures on
M. Then,

dH(Supp(µA), Supp(µB)) ≤ dW,∞(µA, µB).

Proof. See Corollary 2.1 [49].

49

CHAPTER 3

PERSISTENT KÜNNETH THEOREMS

In this chapter, we prove our two Künneth formulae in persistent homology. In section 3.1,
we define two products of (diagrams of) spaces, namely the categorical product in Definition
3.1.1 and the generalized tensor product in Definition 3.1.13, study their properties, and
give examples.
In sections 3.2 and 3.3, we provide the proofs of our persistent Künneth
formulae for the categorical and generalized tensor products, respectively. In section 3.4, as
an application of the categorical Künneth formula for Rips complexes (Corollary 3.2.6), we
revisit the theoretical results about the Rips persistent homology of the N-torus. Parts of
this chapter appeared in [35].

3.1 Products of Diagrams of Spaces

Given a category S of spaces, such as Top, Simp, or Met, and a small indexing category
I, the first objective is to identify relevant products between I-indexed diagrams X ,Y ∈ SI.
For a particular product, the goal is to describe its persistent homology in terms of the
persistent homology of X and Y. This is what we call a persistent Künneth formula. We
begin with the first product construction: the categorical product in SI.

Definition 3.1.1. Let S be a category having all pairwise products (e.g., finitely complete),
let X ,Y ∈ SI, and let X × Y : I −→ S be the functor taking each object i ∈ I to the
(categorical) product Xi × Yi ∈ S, and each morphism i → j in I to the unique morphism
X × Y(i → j) making the following diagram commute:

X (i→j)◦pX
i

Xj

pX
j

Xi × Yi

Xj × Yj

Y(i→j)◦pY
i

Yj

pY
j

(3.1)

where pX

i and pY

j are canonical projections of Xi × Yj onto Xi and Yj respectively. The

50

existence of X × Y(i → j) follows from the universal property defining Xj × Yj.

We have the following observation:

Proposition 3.1.2. Let S be a category with all pairwise products. Then, X × Y ∈ SI is
the categorical product of X ,Y ∈ SI.
Proof. First, note that the existence of pX : X ×Y −→ X and pY : X ×Y −→ Y follows from
: Xi × Yi −→ Yi for each i ∈ I, and the commutativity
that of pX
i
of (3.1) for each i → j.

: Xi × Yi −→ Xi and pY

i

Let Z ∈ SI, and let µ : Z → X , ν : Z → Y be morphisms. For each i ∈ I, let
µi×νi : Zi → Xi×Yi be the unique morphism so that pX
i ◦(µi×νi) = νi.
It readily follows that µ × ν : Z −→ X × Y, given by (µ × ν)(i) = µi × νi, is the unique
morphism of I-indexed diagrams such that pX ◦ (µ × ν) = µ, and pY ◦ (µ × ν) = ν.

i ◦(µi×νi) = µi and pY

Let us describe in more detail the categories we have in mind for S, as well as what the

categorical products are in each case.

Example 3.1.3 (Topological spaces). Recall that Top denotes the category of topological
spaces and continuous maps. For X, Y ∈ Top, their Cartesian product equipped with the
product topology is the categorical product of X and Y in Top.

Example 3.1.4 (Metric Spaces). Let Met denote the category of metric spaces and non-
expansive maps. That is, its objects are pairs (X, dX)—a set and a metric— and its mor-
phisms are functions f : X −→ Y satisfying dY (f(x), f(x0)) ≤ dX(x, x0) for all x, x0 ∈ X.
Definition 3.1.5. Given (X, dX), (Y, dY ) ∈ Met, let X × Y denote the Cartesian product
of the underlying sets, and let dX×Y be the maximum metric on X × Y :

(cid:16)(x, y), (x0, y0)(cid:17) := max{dX(x, x0), dY (y, y0)}.

dX×Y

Since the coordinate projections pX : X × Y −→ X and pY : X × Y −→ Y are non-
expansive, it readily follows that (X × Y, dX×Y ) is the categorical product of (X, dX) and
(Y, dY ) in Met.

51

Example 3.1.6 (Simplicial Complexes). Let Simp be the category of simplicial complexes
and simplicial maps.
Definition 3.1.7. For K, K0 ∈ Simp, let K × K0 be the smallest simplicial complex con-
taining all Cartesian products σ × σ0, for σ ∈ K and σ0 ∈ K0.

In other words, τ ∈ K × K0 if and only if there exist σ ∈ K and σ0 ∈ K0 with τ ⊂ σ × σ0.
Let p : τ → σ and p0 : τ → σ0 be the projection maps onto the first and second coordinate,
respectively. Since p(τ) ⊂ σ, then p(τ) ∈ K, and similarly p0(τ) ∈ K0. Hence p and p0 define
p←− K × K0 p0−→ K0, and it readily follows that K × K0 is the categorical
simplicial maps K
product of K and K0 in Simp.

The Rips complex at scale 
 ∈ R+ is a simplicial complex which can be associated to
any metric space (X, dX). It is widely used in TDA—when (X, dX) is the observed data
set—and it is defined as

(
{x0, . . . , xn} ⊂ X : max
0≤i,j≤n

)
dX(xi, xj) < 


R
(X) :=

.

(3.2)

Notice that any non-expansive function f : X → Y between metric spaces extends to a
simplicial map R
(f) : R
(X) → R
(Y ), and thus R
 defines a functor from Met to Simp.
This functor is in fact compatible with the categorical products:
Lemma 3.1.8. Let (X, dX), (Y, dY ) be metric spaces, and let 
 ∈ R+. Then,

R
(X × Y ) = R
(X) × R
(Y ).

Proof. See the proof of Proposition 10.2 in [1].

One drawback of the categorical product in Simp is that it is not well-behaved with
respect to geometric realizations. Indeed, recall that the geometric realization of a simplicial
complex K—denoted |K|—is the collection of all functions ϕ : K(0) → [0, 1] so that {v ∈

K(0) : ϕ(v) 6= 0} ∈ K, and for which P

ϕ(v) = 1. Moreover, if ϕ, ψ ∈ |K|, then

v∈K(0)

d|K|(ϕ, ψ) := X

|ϕ(v) − ψ(v)|

v∈K(0)

52

defines a metric on |K|. Every simplicial map f : K → L induces a function |f| : |K| → |L|
given by

ϕ(v)

,

|f|(ϕ)(w) = 0 if w /∈ f(K(0))

|f|(ϕ)(w) = X

v∈K(0)
f(v)=w

which, as one can check, is non-expansive. In other words, geometric realization defines a
functor | · | : Simp −→ Met. To see the incompatibility of | · | and the product in Simp,
let K = L = {0, 1,{0, 1}}.
It follows that |K × L| is (homeomorphic to) the geometric

3-simplex n(t0, . . . , t3) ∈ R4 : tj ≥ 0, t0 + ··· + t3 = 1o, while |K| × |L| is the unit square

[0, 1] × [0, 1] ⊂ R2. Hence, |K × L| is in general not equal, nor homeomorphic to |K| × |L|.
This leads one to consider other alternatives; specifically, the category of ordered simplicial
complexes.

Example 3.1.9 (Ordered Simplicial Complexes). If K(0) comes equipped with a partial or-
der so that each simplex is totally ordered, then we say that K is an ordered simplicial com-
plex. A simplicial map between ordered simplicial complexes is said to be order-preserving,
if it is so between 0-simplices. Let oSimp denote the resulting category. For data analysis
applications, and particularly for the Rips complex construction, oSimp is perhaps more
relevant than Simp. Indeed, a data set (X, dX) stored in a computer comes with an explicit
total order on X. If K, K0 ∈ oSimp, with partial orders (cid:22),(cid:22)0, respectively, and σ ∈ K,
σ0 ∈ K0, then the Cartesian product σ×σ0 has a partial order (cid:22)× given by (v, v0) (cid:22)× (w, w0)
if and only if v (cid:22) w and v0 (cid:22)0 w0.

Definition 3.1.10. For K, K0 ∈ oSimp, let K (cid:11) K0 be defined as follows: τ ∈ K (cid:11) K0 if
and only if there exist σ ∈ K and σ0 ∈ K0 so that τ is a totally ordered subset of σ × σ0,
with respect to (cid:22)×.

Just like for simplicial complexes, the coordinate projections p and p0 induce order-

preserving simplicial maps p : K (cid:11) K0 → K and p0 : K (cid:11) K0 → K0. Moreover,

Lemma 3.1.11. Let K and K0 be ordered simplicial complexes. Then,

53

1. K (cid:11) K0 is their categorical product in oSimp.

2. K (cid:11) K0 ⊂ K × K0, where the right hand side is the categorical product in Simp.

Moreover, |K (cid:11) K0| is a deformation retract of |K × K0|.

3. The product map |p| × |p0| : |K (cid:11) K0| −→ |K| × |K0| is a homeomorphism.

Proof. See [31], specifically Definitions 8.1, 8.8 and Lemmas 8.9, 8.11.

The above are the categories of spaces and the products we will consider moving forward.
Going back to diagrams in Top, recall that a CW-complex is a topological space X together
with a CW-structure. That is, a filtration X = {X(k)}k∈N0 of X, so that X(0) has the
discrete topology, X(k) is obtained from X(k−1) by attaching k-dimensional cells Dk ∼=
α ⊂ X via continuous maps ϕα : ∂Dk → X(k−1), and so that the topology on X coincides
ek
with the weak topology induced by X . It follows that X ∈ FTop ⊂ TopN. In this context,
(cid:16)
a cellular map between CW-complexes—i.e. a continuous function f : X → Y so that

X(k)(cid:17) ⊂ Y (k) for all k—is exactly a morphism in TopN. Thus, if CW denotes the

f
category of locally compact CW-complexes (we will see in a moment why this restriction)
and cellular maps, then CW is a full subcategory of FTop.

If X and Y are CW-complexes, then their topological product X × Y can also be written
β with k = i + j,

as a union of cells. Specifically, the k-cells of X × Y are the products ei
for ei

α an i-cell of X, and ej

α × ej

β a j-cell of Y . Thus, if

(X × Y )(k) := [

i+j=k

X(i) × Y (j)

,

k ∈ N0

(3.3)

then n(X × Y )(k)o

will be a CW-structure for X × Y provided the product topology
coincides with the weak topology induced by (3.3). If either X or Y are locally compact,
then this will be the case (see Theorem A.6 in [39]).

k∈N0

In summary: the categorical product in TopN does not recover the usual product of
CW-complexes, which suggests the existence of other useful (non-categorial) products in
FTop and TopN. The product of CW-complexes suggests the following definition,

54

Definition 3.1.12. If X ,Y ∈ FTop, then their tensor product is the filtered space



.

k∈N0

 [

i+j=k

X ⊗ Y :=

Xi × Yj

The tensor product of filtered spaces is sometimes regarded in the literature as the de
facto product filtration; see for instance [8, 11, 64] or [66, Chapter 27]. For general objects in
TopN—e.g., when the internal maps are not inclusions—taking the union of, say, Xi+1 × Yj
and Xi × Yj+1 does not make sense. Instead, the corresponding operation would be to glue
these spaces along the images of the product maps

Xi+1 × Yj ←− Xi × Yj −→ Xi × Yj+1.

Here is where the machinery of homotopy colimits comes in as a means to defining a gener-
alized tensor product in TopN:

Definition 3.1.13. For k ∈ N0, let 4k be the poset category of {(i, j) ∈ N2
with its usual product order, and for X ,Y ∈ TopN, let

0 : i + j ≤ k}

X (cid:2) Y : N2 −→ Top

be the functor sending (i, j) to Xi × Yj, and (i, j) ≤ (s, t) to X (i ≤ s) × Y(j ≤ t). The
generalized tensor product of X and Y is the object X ⊗g Y ∈ TopN given by

X ⊗g Y(k) := hocolim(X (cid:2) Y|4k
)
X (cid:2) Y|4k

X ⊗g Y(k ≤ k0) := hocolim

(cid:18)

(cid:19)

→ X (cid:2) Y|4k0

(3.4)

where the latter is the continuous map associated to the change of indexing categories induced
by the inclusion 4k ⊂ 4k0.

Remark 3.1.14. The definition of the generalized tensor product can be extended to diagrams
indexed by other subcategories of R. For example, to the poset categories of R, R+ (the
set of non negative reals) and R∗ (the set of positive reals). In general, for I = N, R, R+,
or R∗, we let 4r = {(i, j) ∈ I2 : i + j ≤ r} and define X ⊗g Y : I −→ Top as in (3.4).

55

Although 4r and X (cid:2) Y are different for each choice of I, we will use the same notation
for aesthetic purposes. The choice will be clear from the context as we will specify the I in
TopI.

We have the following relation between ⊗ and ⊗g,

Lemma 3.1.15. Let X ,Y ∈ TopN, and let T (X ),T (Y) ∈ FTop be their telescope filtra-
tions. Then, X ⊗g Y is naturally homotopy equivalent to T (X ) ⊗ T (Y).

(cid:16)X ⊗g Y(cid:17)

−→(cid:16)T (X ) ⊗ T (Y)(cid:17)

Proof. We will show that there is a morphism X ⊗g Y −→ T (X ) ⊗ T (Y), so that each map
, k ∈ N0, is a homotopy equivalence. Indeed, since Tj(X )
deformation retracts onto Xj, then each inclusion Xj ,→ Tj(X ) is a homotopy equivalence,
and thus (by Theorem 2.2.9) so is the induced map

k

k

(cid:16)X ⊗g Y(cid:17)

= hocolim

k

(cid:19)

(cid:18)
X (cid:2) Y(cid:12)(cid:12)(cid:12)4k

−→ hocolim

(cid:18)
T (X ) (cid:2) T (Y)(cid:12)(cid:12)(cid:12)4k

(cid:19)

.

Since each inclusion Ti(X ) ,→ Tj(X ) is a closed cofibration (Remark 2.2.12), then the pro-
jection Lemma 2.2.13 implies that the collapse map

(cid:18)

T (X ) (cid:2) T (Y)(cid:12)(cid:12)(cid:12)4k

(cid:19)

hocolim

(cid:18)

T (X ) (cid:2) T (Y)(cid:12)(cid:12)(cid:12)4k

(cid:19)

−→ colim

is a homotopy equivalence. Moreover, since all the maps in T (X )(cid:2)T (Y) are inclusions, and
the colimit of such a diagram is essentially the union of the spaces, then there is a natural
homeomorphism

(cid:18)

T (X ) (cid:2) T (Y)(cid:12)(cid:12)(cid:12)4k

(cid:19) ∼= [

i+j≤k

colim

Ti(X ) × Tj(Y) =(cid:16)T (X ) ⊗ T (Y)(cid:17)

k

which completes the proof.

It follows that,

Corollary 3.1.16. If X ,Y ∈ TopN, then P Hn(X ⊗g Y; F) ∼= P Hn
graded F[t]-modules.

(cid:16)T (X ) ⊗ T (Y); F) as

56

Later on—when studying the persistent singular chains of T (X )⊗T (Y)—it will be useful
to have Proposition 3.1.17 below. The setup is as follows: observe that if X ∈ TopN and
0 < 
 < 1, then

T 

j (X ) := Xj × [j, j + 
) t

Xi × [i, i + 1]

(x, i + 1) ∼ (fi(x), i + 1)

  G

i<j

!(cid:30)

is an open neighborhood of Tj(X ) in Tj+1(X ), which deformation retracts onto Tj(X ).
Indeed, any element x ∈ T 

j (X ) can be written uniquely as x = (x, t) where either x ∈ Xj
and t ∈ [j, j + 
), or x ∈ Xi for i < j and t ∈ [i, i+1). In these coordinates Tj(X ) = {(x, t) ∈
T 

j (X ) : t ≤ j}, and a deformation retraction hX
j (X ) can be defined
as

j (X ) × [0, 1] → T 


: T 


j

hX
j

(cid:16)

(x, t)

t ≤ j


(cid:16)(x, t), λ
(cid:17) =
j (X )(cid:17). Then,
i (X ) and (i ≤ j) 7→(cid:16)T 

deformation retracts onto (cid:16)T (X ) ⊗ T (Y)(cid:17)
k (X ) deformation retracts onto

(cid:17)
x, λj + (1 − λ)t
i (X ) ,→ T 


t ≥ j

(3.5)

if

if

.

k

Proposition 3.1.17. If X ,Y ∈ TopN and k ∈ N0, then T 


Tk(X ), and (cid:16)T 
 (X ) ⊗ T 
 (Y)(cid:17)

k

Let T 
 (X ) ∈ FTop be i 7→ T 


Proof. Given k ∈ N0, our goal is to define a deformation retraction

Hk : [

i+j=k

j (Y) × [0, 1] −→ [

T 

i (X ) × T 


T 

i (X ) × T 


j (Y) .

i+j=k

To this end, we subdivide (T 
 (X ) × T 
 (Y))k as follows: For i + j = k, let

Ak = (T (X ) × T (Y))k
Bi,j = Xi × [i, i + 
) × Yj × [j, j + 
)

Ci,j−1 = Xi × (i, i + 
) × Yj−1 × [j − 1 + 
, j)
Di,j−1 = Xi × [i + 
, i + 1) × Yj−1 × (j − 1, j − 1 + 
)
Ei,j−1 = Xi × (i, i + 
) × Yj−1 × (j − 1, j − 1 + 
)

57

i,j−1 where (cid:16)(x, t), (y, s)(cid:17) ∈ E+

We further write Ei,j−1 = E+
i,j−1) if
and only if t − i ≥ s + 1 − j (sim. t − i ≤ s + 1 − j). Figure 3.1 below visually represents
the subsets defined above for k = 3.

i,j−1 (sim. E−

i,j−1 ∪ E−

Figure 3.1: Visual representation for the deformation retract H3.

Let us now define the map Hk. Let Hk : Ak × [0, 1] −→ Ak be the projection onto the

i (x, λ), hY
hX

Hk

i , hY

where hX

first coordinate. On each Bi,j × [0, 1] we define
(cid:17) =(cid:16)

(cid:16)(x, y), λ

j are given by (3.5). For elements in (cid:16)
(cid:18)

(cid:18)
y, (1 − λ)s + λ

(cid:17) =(cid:16)(x, t), (y, s), λ
(cid:16)x, y, λ
(cid:17) to
(cid:18)
And finally, for elements in (cid:16)

i (x, λ),
hX

j − j − s
1 + i − t
!!

(cid:17) × [0, 1], Hk takes (x, y, λ) to
Di,j−1 ∪ E−
 
i,j−1
i − 1 + i − t
j − s

!
j−1(y, λ)

, hY

x, (1 − λ)t + λ

  

j (y, λ)(cid:17)

i,j−1

Ci,j−1 ∪ E+
(cid:19)(cid:19)(cid:19)

(cid:17) × [0, 1], Hk takes

Continuity readily follows by construction. Furthermore, the marginal function Hk(−, 0) is
the identity everywhere and Hk(−, 1) ∈ Ak. This finishes the proof.

3.1.1 A comparison theorem

Next we show that the persistent homology of the categorical and generalized tensor products
are interleaved in the log scale. While a full characterization of stability for each product fil-

58

tration is beyond the scope of this dissertation, this section showcases some of the techniques
one can employ when addressing this question.

Definition 3.1.18. For δ ∈ R, the δ-shift functor Tδ : CR → CR is defined as follows: For
M ∈ CR, let Tδ(M) ∈ CR be the functor sending r ∈ R to M(r + δ), and r ≤ r0 ∈ R to
M(r+δ ≤ r0+δ). For a morphism ϕ : M → N , we get a morphism Tδ(ϕ) : Tδ(M) → Tδ(N ),
and if δ ≥ 0, then there is a morphism TM
(cid:17)

: M → Tδ(M) satisfying

(cid:16)

δ

TM

δ

r

= M (r ≤ r + δ)

for all r ∈ R. We call TM

δ

the δ-transition morphism.

The notion of interleavings, first introduced in [18], allows one to compare objects in

ModRF . Specifically:

Definition 3.1.19. Let M,N ∈ CR and δ ≥ 0. A δ-interleaving between M and N
consists of morphisms ϕ : M → Tδ(N ) and ψ : N → Tδ(M) so that Tδ(ϕ) ◦ ψ = TN
2δ and
Tδ(ψ) ◦ ϕ = TM

2δ . We define the interleaving distance dI between M and N as:

dI (M,N ) := infn

δ | M and N are δ-interleaved o

.

If there are no interleavings between M and N , we say that dI(M,N ) = ∞.

Let R∗ denote the poset category associated to the set of positive reals, R∗, and define

the logarithm functor ln : TopR∗ → TopR as

ln(X )(r) = X (er)

ln(X )(r ≤ r0) = X (er ≤ er0

).

Then,

Theorem 3.1.20. If X ,Y ∈ TopR∗, then

(cid:18)

dI

Hn

(cid:16)ln(X × Y); F(cid:17)

(cid:16)ln(X ⊗g Y); F(cid:17)(cid:19)

≤ ln(2).

, Hn

59

Proof. For each r ∈ R∗, let (cid:3)r = {(i, j) ∈ R2∗ : max{i, j} ≤ r}. We have inclusions
4r ,→ (cid:3)r ,→ (cid:3)2r and (cid:3)r ,→ 42r, and thus by Proposition 2.2.8, the triangles in the
diagram

hocolim(X (cid:2) Y|42r

)

hocolim(X (cid:2) Y|48r

)

(3.6)

hocolim(X (cid:2) Y|(cid:3)r)

hocolim(X (cid:2) Y|(cid:3)4r

)

commute. Since (cid:3)r has a terminal object, namely (r, r), then the natural map

hocolim(X (cid:2) Y(cid:12)(cid:12)(cid:12)(cid:3)r

) −→ Xr × Yr

is a weak homotopy equivalence (Theorem 2.2.10). Such maps induce isomorphisms at the
level of homology [39, Theorem 4.21], and thus applying ln followed by Hn( · ; F) turns (3.6)
into a ln(2)-interleaving of the desired objects in ModRF .

If M,N ∈ ModRF are pointwise finite, then—in addition to their interleaving distance
dI(M,N )—one can compute the bottleneck distance dB between bcd(M) and bcd(N ) as
follows. A matching between two multisets M and N is a multiset bijection ϕ : SM −→ SN
between some SM ⊂ M and SN ⊂ N. Given ϕ, we say that I ∈ SM and ϕ(I) ∈ SN are
matched, and all other elements of (M (cid:114) SM) ∪ (N (cid:114) SN) are called unmatched. Let δ > 0.
A matching between bcd(M) and bcd(N ) is called a δ-matching if the following conditions
hold:

1. If I and J are matched, then

max{|‘I − ‘J|,|ρI − ρJ|} ≤ δ.

Recall that ‘I and ρI are, respectively, the left and right endpoints of the interval I.

2. If I is unmatched, then |‘I − ρI| ≤ 2δ.
The bottleneck distance dB between bcd(M) and bcd(N ) is defined as:
(cid:16)bcd(M), bcd(N )(cid:17) := infn

δ | bcd(M) and bcd(N ) are δ-matched o

dB

60

If no δ-matchings exist, then the bottleneck distance is ∞.

The Isometry Theorem [7] implies that if M,N ∈ ModRF are pointwise finite, then

dI(M,N ) = dB (bcd(M), bcd(N )) .

The following corollary is a direct consequence of the above equality and Theorem 3.1.20.

Corollary 3.1.21. Let X ,Y ∈ TopR∗ be so that X × Y and X ⊗g Y are pointwise finite.
Then

(cid:16)bcdn(X × Y), bcdn(X ⊗g Y)(cid:17)—which exists by [19,

1. If under a matching realizing dB

Theorem 5.12]—we have that I is matched to J, then

1
2 ≤ ‘I

‘J

,

ρI
ρJ

≤ 2.

2. If I is unmatched, then ρI
‘I

≤ 4.

3.2 A Persistent Künneth formula for the categorical product

Let I be a small indexing category and let S be any of the categories of spaces from
Section 3.1 (that is S = Top, Met, Simp, or oSimp). In this section, we relate the rank
invariants of two objects in SI to that of their categorical product, and prove the persistent
Künneth formula for the categorical product in SP, where P is the poset category of a
separable totally ordered poset P.

Remark 3.2.1. Recall that the topological Künneth formula (Corollary 2.2.5) holds for objects
X, Y ∈ Top and singular homology. Every (X, dX) ∈ Met can be viewed as a topological
space via the metric topology, and since the maximum metric induces the product topology,
then the Künneth formula holds for Met. Let K, L ∈ oSimp, and let | · | : oSimp → Top
be the geometric realization functor. From the homeomorphism |K (cid:11) L| → |K| × |L| and
the natural isomorphism Hn(|K|; R) ∼= Hn(K; R) between singular and simplicial homology,
the formula holds in oSimp. Using (2) of lemma 3.1.11, the formula also holds for Simp.

61

Let F be a field and let X and Y be objects in SI. Using the topological Künneth formula
and the observation that Hn(Xr; F) and Hn(Yr; F) are vector spaces for each n ∈ N0, r ∈ I,
we obtain isomorphisms

M

i+j=n

Hi(Xr; F) ⊗F Hj(Yr; F)

∼=−−−→ Hn(Xr × Yr; F).

Using naturality, this yields an isomorphism in ModIF between Hn(X ×Y; F) and Mn, where

Mn(r) := M

i+j=n

Hi(Xr; F) ⊗F Hj(Yr; F)

(3.7)

and the homomorphism Mn(r → r0) is the sum of the ones induced between tensor products
by the maps Xr → Xr0 and Yr → Yr0. In other words, and using the notation of tensor
product and direct sum of objects in ModIF (see Definition 2.3.7), we obtain the following:
Lemma 3.2.2. If X ,Y ∈ SI, then for every n ∈ N0 and every field F

Hn(X × Y; F) ∼= M

i+j=n

Hi(X ; F) ⊗ Hj(Y; F)

in ModIF.

An immediate consequence is the following calculation at the level of rank invariants (see

Definition 2.3.22):

Proposition 3.2.3. Let X ,Y ∈ SI and let ρX , ρY be their rank invariants. Then

(r → r0) = X

ρX×Y

n

i+j=n

i (r → r0) · ρY
ρX

j (r → r0)

for every r → r0 in I, with the convention that ∞ · 0 = 0 = 0 · ∞.

Proof. This follows by direct inspection of the homomorphism

fr0,r ⊗F gr0,r : Hi(Xr; F) ⊗F Hj(Yr; F) −→ Hi(Xr0; F) ⊗F Hj(Yr0; F)

a ⊗F b 7→ fr0,r(a) ⊗F gr0,r(b)

62

for each r → r0 ∈ I. Indeed, since Img(fr0,r ⊗F gr0,r) = Img(fr0,r) ⊗F Img(gr0,r), then the
result follows from the fact that the dimension of the tensor product of two vector spaces
equals the product of the dimensions.

We are now ready to prove the main result of this section: the Künneth formula for the

categorical product X × Y.

Theorem 3.2.4. Let P be the poset category of a separable (with respect to the order topol-
ogy) totally ordered set. Let X ,Y ∈ SP be P-indexed diagrams of spaces, and assume that
Hi(X ; F) and Hj(Y; F) are pointwise finite for each 0 ≤ i, j ≤ n. Then Hn(X × Y; F) is
pointwise finite, and its barcode satisfies:

bcdn (X × Y; F) = [

i+j=n

(cid:26)

I ∩ J

(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ; F), J ∈ bcdj(Y; F)
(cid:27)

where the union on the right is of multisets (i.e., repetitions may occur).

Proof. The fact that Hn(X × Y; F) is pointwise finite follows directly from Lemma 3.2.2. As
for the barcode formula, and removing the field F from the notation, we have the following
sequence of isomorphisms

i+j=n

Hn(X × Y) ∼= M
∼= M
∼= M
∼= M

i+j=n

i+j=n

i+j=n

 ⊗

 M

J∈bcdj(Y)



1J

1I

Hi(X ) ⊗ Hj(Y)

 M
I∈bcdi(X )
M
I∈bcdi(X )
J∈bcdj(Y)
M
I∈bcdi(X )
J∈bcdj(Y)

1I ⊗ 1J

1I∩J .

The direct sum distributes with respect to the tensor product in ModPF since it does so in
ModF. The last isomorphism is the one from Example 2.3.8, and the theorem follows from
the uniqueness of the barcode.

63

This result generalizes to the product of k objects in SP by inductively using bcdj (Xk)

and bcdi (X1 × ··· × Xk−1) to compute bcdi+j (X1 × ··· × Xk):
Corollary 3.2.5. Let X1, . . . ,Xk ∈ SP. Assume that for each 1 ≤ j ≤ k, 0 ≤ nj ≤ n,
Hnj(Xj) is pointwise finite. Then
bcdn (X1 × ··· × Xk) =

(
I1 ∩ ··· ∩ Ik

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ij ∈ bcdnj(Xj),

)
nj = n

.

kX

j=1

The result which is perhaps most relevant to persistent homology computations with

Rips complexes is as follows:
Corollary 3.2.6. Let (X, dX), (Y, dY ) be finite metric spaces and let bcdR
barcode of the Rips filtration R(X, dX) := {R
(X, dX)}
≥0. Then,

n (X × Y, dX×Y ) = [

bcdR

i+j=n

(cid:26)

I ∩ J

(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdR

i (X, dX) , J ∈ bcdR

(cid:27)
j (Y, dY )

n (X, dX) be the

for all n ∈ N0, if dX×Y is the maximum metric.

Proof. Fix total orders on X and Y . Then, their 
-Rips complexes are ordered simplicial
complexes, and by lemmas 3.1.8 and 3.1.11 we have that:

|R
(X × Y )| = |R
(X) × R
(Y )| ’ |R
(X) (cid:11) R
(Y )| ∼= |R
(X)| × |R
(Y )|.

Since the equivalences commute with inclusions of Rips complexes, the result follows.

3.3 A Persistent Künneth Formula for the generalized tensor prod-

uct

In this section, we will establish the Künneth formula for the generalized tensor product
X ⊗g Y of two objects X ,Y ∈ TopN. We start with a brief discussion on the grading of the
tensor product and Tor of two graded modules over a graded ring R. The discussion then
specializes to R = F[t] in order to establish the F[t]-isomorphisms

(cid:17) ⊗F[t]

(cid:16)

P 1

I=[‘I ,ρI)

P 1

J=[‘J ,ρJ )

(cid:16)

(cid:16)

TorF[t]

P 1

I=[‘I ,ρI) , P 1

J=[‘J ,ρJ )

64

(cid:17) ∼= P 1(‘J +I)∩(‘I+J)
(cid:17) ∼= P 1(ρJ +I)∩(ρI+J)

(3.8)

for intervals I, J ⊂ N0. We end with a persistent Eilenberg-Zilber theorem, from which the
aforementioned persistent Künneth theorem will follow.

3.3.1 The tensor product and Tor of graded modules

Recall that a commutative ring R with unity is called graded if it can be written as R =
R0 ⊕ R1 ⊕ ··· where each Ri is an additive subgroup of R and RiRj ⊂ Ri+j for every
i, j ∈ N0. Similarly, an R-module A is graded if it can be written as A = A0 ⊕ A1 ⊕ ··· ,
where the Ai’s are subgroups of A and RiAj ⊂ Ai+j for all i, j ∈ N0. An element a ∈ A
(resp. in R) is called homogeneous of degree i ∈ N0 if a ∈ Ai (resp. in Ri), and we will use
the notation deg(a) = i.

Let A, B be graded modules over a graded ring R. Our first goal is to describe a direct
sum decomposition of the R-module A ⊗R B inducing the structure of a graded R-module.
Indeed, since Ai and Bj are R0-modules for all i, j ∈ N0, then so are A, B and we have the
R0-isomorphism

A ⊗R0 B ∼= M

 M

k∈N0

i+j=k

 .

Ai ⊗R0 Bj

Fix k ∈ N0 and let Jk be the R0-submodule of A ⊗R0 B generated by elements of the
form (ra) ⊗ b − a ⊗ (rb), where a ∈ A, b ∈ B and r ∈ R are homogeneous elements with
deg(a) + deg(r) + deg(b) = k. It follows that Jk is a submodule of

(cid:16)

(cid:17)

k

A ⊗R0 B

:= M

i+j=k

Ai ⊗R0 Bj

and thus Jk ∩ J‘ = {0} if k 6= ‘. Let J = J0 ⊕ J1 ⊕ ··· . Any homogeneous element r ∈ R
of degree ‘ induces a well-defined R0-homomorphism

(A ⊗R0 B)k/Jk −→ (A ⊗R0 B)k+‘/Jk+‘
a ⊗ b + Jk

7→ (ra) ⊗ b + Jk+‘ = a ⊗ (rb) + Jk+‘

and therefore

A ⊗R0 B/J ∼= M

k∈N0

(A ⊗R0 B)k/Jk

65

(3.9)

inherits the structure of a graded R-module.

The inclusion R0 ,→ R induces an R0-epimorphism ι : A ⊗R0 B −→ A ⊗R B with
ker(ι) = J. Indeed, the elements of J are exactly the relations missing between A ⊗R0 B
and A ⊗R B when defining the latter as a quotient module. It follows that ι induces an
isomorphism

ι∗ : A ⊗R0 B/J −→ A ⊗R B

of R-modules, and the grading on the codomain induced by (3.9) turns ι∗ into a graded
isomorphism of graded R-modules. We summarize this analysis as follows:

Proposition 3.3.1. Let A, B be graded modules over a graded ring R. Then A ⊗R B
decomposes as the direct sum of the subgroups

(cid:12)(cid:12)(cid:12)(cid:12) aα ∈ A, bα ∈ B homogeneous, deg(aα) + deg(bα) = k

)

(A ⊗R B)k =

aα ⊗ bα

(X

α

and R‘ · (A ⊗R B)k ⊂ (A ⊗R B)k+‘ for all k, ‘ ∈ N0. In other words, A ⊗R B is a graded
R-module.

And now we prove the corresponding proposition for TorR(A, B):

Proposition 3.3.2. If A, B are graded R-modules, then so is TorR(A, B).

Proof. Fix a graded free resolution

··· → F2(A) → F1(A) → F0(A) → F−1(A) = A → 0

defined inductively as follows. Let F0(A) be the free R-module generated by the homoge-
neous elements of A, and define for each i ∈ N0 the additive subgroup

)
(cid:12)(cid:12)(cid:12)(cid:12) rα ∈ R, aα ∈ A homogeneous with deg(rα) + deg(aα) = i

.

F0,i(A) =

rα · aα

(X

α

Then F0(A) = F0,0(A) ⊕ F0,1(A) ⊕ ··· , and with this decomposition, the natural map
F0(A) → F−1(A) is a surjective graded homomorphism of graded R-modules. In particular,
the kernel of this homomorphism is a graded R-submodule of F0(A). We proceed inductively

66

by letting Fj+1(A) be the graded free R-module generated by the homogeneous elements in
the kernel of Fj(A) → Fj−1(A).

Taking the tensor product with B over R yields

··· → F2(A) ⊗R B → F1(A) ⊗R B → F0(A) ⊗R B → A ⊗R B → 0

and we note that:

1. Each Fj(A)⊗R B is a graded R-module, by Proposition 3.3.1, and each Fj(A)⊗R B →

Fj−1(A) ⊗R B is a graded R-homomorphism.
(cid:18)

Fj(A) ⊗R B → Fj−1(A) ⊗R B

2. ker

(cid:17)

(cid:19)
(cid:18)(cid:16)Fj(A) ⊗R B

k

ker

M
(cid:18)
k∈N0
Fj+1(A) ⊗R B → Fj(A) ⊗R B
M
k∈N0

(cid:18)(cid:16)Fj+1(A) ⊗R B

Img

(cid:17)

k

decomposes as the direct sum

→(cid:16)Fj−1(A) ⊗R B

(cid:17)

(cid:19)

k

(cid:19)
→(cid:16)Fj(A) ⊗R B

decomposes as

(cid:17)

(cid:19)

k

and similarly, Img

with the image being a graded R-submodule of the kernel.

3. TorR(A, B) inherits the structure of a graded R-module from the decomposition

TorR(A, B) ∼= M

ker

(cid:18)(cid:16)F1(A) ⊗R B
(cid:17)
(cid:18)(cid:16)F2(A) ⊗R B
(cid:17)

k

k

(cid:17)
→(cid:16)F0(A) ⊗R B
→(cid:16)F1(A) ⊗R B
(cid:17)

(cid:19)
(cid:19)

k

k

k∈N0

Img

finishing the proof.

We now specialize to the case R = F[t] and present explicit computations for interval
modules. For m ∈ N0, k ∈ N0 ∪ {∞}, let Im,k denote the graded F[t]-module P 1[m,m+k).
For brevity, we will drop the subscript k if it is ∞. That is, we will use Im to denote
P 1[m,∞). Then, Im,k
Proposition 3.3.3. Let m, n ∈ N0, k, l ∈ N0 ∪ {∞}. Then Im,k ⊗F[t] In,l
and TorF[t](Im,k,In,l) ∼= Im+n+max{k,l},min{k,l}.

∼= tmF[t].(tm+k) and Im ∼= tmF[t].

∼= Im+n,min{k,l}

67

Proof. There is a graded F[t]-isomorphism Im ⊗F[t] In → Im+n, which takes the generator
tm ⊗F[t] tn to tm+n. Since Im,k and In,l are quotient modules of Im and In respectively,
their tensor product is isomorphic to a quotient module of Im+n, say Im+n,q for some
gives tp ·(cid:16)
appropriate 1 ≤ q ≤ ∞. The isomorphism takes tm ⊗F[t] tn to tm+n and the action of tp
or if tp+n is zero in In,l. This implies that if p ≥ min{k, l}, then tm+n+p is zero in Im+n,q.
The minimum of all such p is min{k, l} and therefore, q = min{k, l}.
To compute TorF[t]

tm ⊗F[t] tn(cid:17) → tm+n+p. The left hand side is zero if either tp+m is zero in Im,k

(cid:17), we fix a graded free resolution for Im,k:

(cid:16)Im,k,In,l

0 → Im+k → Im → Im,k → 0.
Taking the tensor product of the sequence with In,l gives us:

Im+k ⊗F[t] In,l → Im ⊗F[t] In,l → Im,k ⊗F[t] In,l → 0.
Using the definition of Tor and the tensor product of intervals, we get that
TorF[t](Im,k,In,l) ∼= Ker
Im+k ⊗F[t] In,l → Im ⊗F[t] In,l
∼= Ker
Im+k+n,l → Im+n,l
∼= Im+n+max{k,l},min{k,l}

(cid:18)
(cid:18)

(cid:19)

(cid:19)

where the last isomorphism is justified by the following argument: the element tm+p+n ∈
Im+k+n,l goes to tptm+n ∈ Im+n,l where p ≥ k. The latter is zero if and only if p ≥ l, or
said equivalently p ≥ max{k, l}. This finishes the proof.

Remark 3.3.4. This last proposition recovers the formulas shown in (3.8). Moreover, using
bilinearity, these isomorphisms can be used to compute the tensor product and Tor of any
pair of finitely generated graded F[t]-modules.

3.3.2 The Graded Algebraic Künneth Formula

The Algebraic Künneth Formula for graded modules over a graded PID R is proved in
exactly the same way as Theorem 2.2.3, by noticing that all the steps can be carried out in

68

a degree-preserving fashion. A few versions of this have appeared recently, see for instance
[62, Proposition 2.9] or [12, Theorem 10.1], and we include it next for completeness:
Theorem 3.3.5. Let C and C0 be two chain complexes of graded modules over a graded PID
R, and assume that one of C, C0 is flat. Then, for each n ∈ N0, there is a natural short
exact sequence

0 −→ M

i+j=n

Hi(C) ⊗R Hj(C0)

µ−−→ Hn(C ⊗R C0)

ν−−→

M

TorR(Hi(C), Hj−1(C0)) −→ 0

which splits, though not naturally.

i+j=n

It is at this point that we depart from the existing literature on persistent Künneth
formulas for the tensor product of filtered chain complexes. Indeed, next we will turn these
algebraic results into theorems at the level of filtered topological spaces. We begin with the
following result:
Theorem 3.3.6. Let X ,Y ∈ TopN. Then, for all n ∈ N0, we have a natural short exact
sequence

0 → M

i+j=n

P Hi(X ) ⊗F[t] P Hj(Y) → Hn

(cid:16)

P S∗(T (X )) ⊗F[t] P S∗(T (Y))(cid:17) →
M

P Hi(X ), P Hj−1(Y)(cid:17) → 0

TorF[t]

(cid:16)

i+j=n

which splits, but not naturally.
Proof. Since T (X ) and T (Y) are filtered spaces, then P S∗(T (X )) and P S∗(T (Y)) are chain
complexes of free graded F[t]-modules, and hence flat. Now, each inclusion ιi : Xi ,→ Ti(X )
is a homotopy equivalence, and thus induces an isomorphism at the level of homology. One
thing to note is that if i < j, then

Ti(X )
ιi

Tj(X )
ιj

Xi

X (i<j)

Xj

69

commutes only up to homotopy, but this is enough to conclude that ι = {ιi}i∈N0 induces
natural isomorphisms ι∗ : Hn(X ; F) −→ Hn(T (X ); F), n ∈ N0. The result follows from
plugging C = P S∗(T (X )) and C0 = P S∗(T (Y)) into Theorem 3.3.5, and using the natural
F[t]-isomorphisms P ι∗ : P Hi(X ; F) −→ P Hi(T (X ); F).

3.3.3 A Persistent Eilenberg-Zilber theorem
Our next objective is to show that the chain complexes P S∗(T (X )) ⊗F[t] P S∗(T (Y)) and
P S∗(X ⊗g Y) have naturally isomorphic homology. The result follows, as we will show in
Theorem 3.3.11, from applying the machinery of acyclic models [30] to the functors

E : FTop0 × FTop0 −→ gChF[t]

(X ,Y) 7→ P S∗(X ) ⊗F[t] P S∗(Y)

F : FTop0 × FTop0 −→ gChF[t]

(X ,Y) 7→ P S∗(X ⊗ Y)

Here FTop0 is the full subcategory of FTop comprised of those filtered spaces X = {Xj}j∈N0
where Xj is an open subset of Xj+1 for all j ∈ N0, and P Sn(X ⊗Y) is the graded F[t]-module
whose degree-k component is the sum of vector spaces

Sn(Xk × Y0; F) + Sn(Xk−1 × Y1; F) + ··· + Sn(X1 × Yk−1; F) + Sn(X0 × Yk; F),

(cid:16)(X ⊗Y)k; F(cid:17). The models

where each Sn(Xk−‘×Y‘; F) is regarded as a linear subspace of Sn
in question are a family M of objects from FTop0 × FTop0 satisfying certain properties
(Lemmas 3.3.7 and 3.3.8) implying our Persistent Eilenberg-Zilber Theorem 3.3.10. We start
by defining M.

For X ∈ Top, let ΣeX ∈ FTop0 be the filtered space with the empty set ∅ at indices

0 ≤ i < e, and X for i ≥ e:

ΣeX : ∅ ⊂ ··· ⊂ ∅ ⊂ X ⊂ X ⊂ ···

70

Let ∆p ⊂ Rp+1 be the standard geometric p-simplex, and let M be the set of objects
(Σe∆p, Σf∆q) in FTop0 × FTop0 for p, q, e, f ∈ N0. We have the following:
Lemma 3.3.7. Each M ∈ M is both E-acyclic and F-acyclic. That is, the homology groups
Hn(E(M)) and Hn(F(M)) are trivial for all n > 0.

Proof. If M =(cid:16)Σe∆p, Σf∆q(cid:17), then

(cid:16)Σe∆p ⊗ Σf∆q(cid:17) ∼= P S∗(Σe+f∆p × ∆q)

F (M) = P S∗

as chain complexes of graded F[t]-modules. Since ∆p × ∆q is convex, then

Hn(F(M)) ∼= P Hn(Σe+f∆p × ∆q) = 0

for all n > 0, showing that M is F-acyclic.

To show that M is E-acyclic, notice that P Hn (Σe∆p) = 0 for n > 0, and that
P H0 (Σe∆p) ∼= teF[t] is a free graded F[t]-module. By the graded algebraic Künneth Theo-
rem 3.3.5, with C = P S∗(Σe∆p) and C0 = P S∗(Σf∆q), we get that

Hn(E(M)) = Hn(P S∗(Σe∆p) ⊗F[t] P S∗(Σf∆q)) = 0

for all n > 0, since the tensor products and Tor modules in the short exact sequence are all
zero.

Lemma 3.3.8. For each n ∈ N0, the functors Fn and En are free with base in M.
Proof. Let n ∈ N0. The functor Fn (resp. En) being free with base in M means that for
all (X ,Y) ∈ FTop0 × FTop0, the graded F[t]-module Fn(X ,Y) (resp. En(X ,Y)) is freely
generated over F[t] by some subset of

[

Img

M∈M

u:M→(X ,Y)

(cid:18)

(cid:19)
Fn(u) : Fn(M) −→ Fn(X ,Y)

where u ranges over all morphisms in FTop0 × FTop0 from M ∈ M to (X ,Y).

71

Starting with F, we will first construct a free F[t]-basis for

Fn(X ,Y) = P Sn(X ⊗ Y) = M

X

Sn(Xi × Yj)

k∈N0

i+j=k

consisting of singular n-simplices on the spaces Xi × Yj. We proceed inductively as follows:
Let S0 be the collection of all singular n-simplices ∆n → X0 × Y0, and assume that for a
fixed k ≥ 1 and every 0 ≤ ‘ < k, we have constructed a set S‘ of simplices ∆n → Xp × Yq,
p + q = ‘, so that

Ak−1 = tk−1S0 ∪ tk−2S1 ∪ ··· ∪ tSk−2 ∪ Sk−1

is a basis, over F, for

P Sn(X ⊗ Y)k−1 = X

p+q=k−1

Sn(Xp × Xq).

Since P Sn(X ⊗ Y) is an F[t]-submodule of P Sn(X ⊗ Y) and the latter is torsion free, then
tAk−1 is an F-linearly independent subset of P Sn(X ⊗ Y)k. We claim that tAk−1 can
be completed to an F-basis Ak of P Sn(X ⊗ Y)k, consisting of singular n-simplices ∆n →
Xi × Yj, i + j = k. Indeed, if C is the collection of all sets C of simplices ∆n → Xi × Yj,
i + j = k, so that C is F-linearly independent in P Sn(X ⊗ Y)k and tAk−1 ⊂ C, then C can
be ordered by inclusion, and any chain in C is bounded above by its union. By Zorn’s lemma
C has a maximal element Ak, which yields the desired basis. If we let Sk = Ak (cid:114) tAk−1,
then

S := [

k∈N0

Sk

is a free F[t]-basis for P Sn(X ⊗ Y).

morphism u : M → (X ,Y) in FTop2
some i, j ∈ N0, then there are unique singular simplices σX

We claim that each σ ∈ S is in the image of Fn(u) for some model M ∈ M and some
0. Indeed, since σ is of the form σ : ∆n → Xi × Yj for
(cid:17) ◦ d. Let
: ∆n → Yj,
: Σi∆n → X be the morphism in FTop0 defined as the inclusion ∅ ,→ X‘ for ‘ < i,

so that if d : ∆n → ∆n × ∆n is the diagonal map d(a) = (a, a), then σ =(cid:16)
: ∆n → Xi and σY
σX
i , σY
j

j

i

ΣσX
i

72

fashion. Then, (cid:16)ΣσX

and as the composition ∆n
i , ΣσY

j

di,j ∈ Sn

σX
i−−→ Xi ,→ X‘ for ‘ ≥ i. Define ΣσY

(cid:17) :(cid:16)Σi∆n, Σj∆n(cid:17) −→ (X ,Y) is a morphism in FTop2
(cid:16)(Σi∆n)i × (Σj∆n)j

j

: Σj∆n → Y in a similar

0, and if

is the singular n-simplex corresponding to the diagonal map d, then

(cid:17) = Sn(∆n × ∆n)
(cid:17) (di,j) = σ.

(cid:16)ΣσX

Fn

i , ΣσY
j

As for the freeness of

En(X ,Y) = M

p+q=n

P Sp(X ) ⊗F[t] P Sq(Y),

we note that each direct summand P Sp(X ) ⊗F[t] P Sq(Y) is freely generated over F[t] by
elements which can be written as the tensor product of a homogeneous element from P Sp(X )
and a homogeneous element from P Sq(Y). That is, by elements of the form σX
p ⊗F[t] σY
q ,
p : Σe∆p → X
p : ∆p → Xe and σY
where σX
q : Σf∆q −→ Y be defined as in the analysis of Fn above. Let ip ∈ P Sp(Σe∆p)
and Σf σY
be the homogeneous element of degree e corresponding to the identity of ∆p, and define
iq ∈ P Sq(Σf∆q) in a similar fashion. Then, ip ⊗F[t] iq ∈ En(Σe∆p, Σf∆q) and

q : ∆q → Yf are singular simplices. We let ΣeσX

(cid:16)ΣeσX

p , Σf σY
q

(cid:17) (ip ⊗F[t] iq),

p ⊗F[t] σY
σX

q = En

thus completing the proof.

Lemma 3.3.9. The functors H0F and H0E are naturally equivalent.
Proof. Given (X ,Y) ∈ FTop0 × FTop0, our goal is to define a natural isomorphism

(cid:16)

P S∗(X ) ⊗F[t] P S∗(Y)(cid:17) −→ H0

(cid:16)

P S∗(X ⊗ Y)(cid:17)

.

Ψ : H0

The first thing to note is that the graded algebraic Künneth formula (Theorem 3.3.5) yields
a natural isomorphism

(cid:16)
P S∗(X ) ⊗F[t] P S∗(Y)(cid:17) ∼= P H0(X ) ⊗F[t] P H0(Y).

H0

73

Moreover, since each Xk−‘ × Y‘ is open in (X ⊗ Y)k, then the inclusion

X

i+j=k

Sn(Xi × Yj) ,→ Sn((X ⊗ Y)k)

is a chain homotopy equivalence (see [39, Proposition 2.21]), and thus

(cid:16)

P S∗(X ⊗ Y)(cid:17) ∼= P Hn(X ⊗ Y).

Hn

Therefore, it is enough to construct a natural isomorphism

Ψ : P H0(X ) ⊗F[t] P H0(Y) −→ P H0(X ⊗ Y)

and we will do so on each degree k as follows. Recall that (see Proposition 3.3.1)

(cid:16)

P H0(X ) ⊗F[t] P H0(Y)(cid:17)
where Jk is the linear subspace of L

k

i+j=k

 M

i+j=k

∼=

form (t‘a) ⊗F b − a ⊗F (t‘b) with deg(a) + deg(b) + ‘ = k. Let

(cid:30)

Jk

H0(Xi) ⊗F H0(Yj)

H0(Xi) ⊗F H0(Yj) generated by elements of the

ψk : L

i+j=k

H0(Xi) ⊗F H0(Yj) −→ H0((X ⊗ Y)k)

i ⊗F ΓY
ΓX

j

7→ Γ

(resp. ΓY

j ) is a path-connected component of Xi (resp. Yj), i+ j = k, and Γ is the
where ΓX
i
unique path-connected component of (X ⊗ Y)k containing ΓX
j . This definition on basis
elements uniquely determines ψk as an F-linear map, and we claim that it is surjective with
ker(ψk) = Jk. Surjectivity follows from observing that if (x, y) ∈ Γ, then (x, y) ∈ Xi × Yj for
some i + j = k, and that if ΓX
j are the path-connected components in Xi and Yj,
respectively, so that (x, y) ∈ ΓX
j and maximality
of Γ imply ΓX

j , then path-connectedness of ΓX

i and ΓY
i ×ΓY

i ×ΓY

i ×ΓY

i × ΓY

j ⊂ Γ.

The inclusion Jk ⊂ ker(ψk) is immediate, since it holds true for elements of the form
k−j ⊗F ΓY

(cid:17), and these generate Jk over F. In order to show that

(cid:16)
t‘ΓX
ker(ψk) ⊂ Jk, we will first establish the following:

k−j ⊗F t‘ΓY
j−‘

j−‘ − ΓX

74

i × ΓY

Claim 1: If ΓX
p + q = i + j = k, such that ψk

(cid:16)ΓX
j ⊂ Xi × Yj and ΓX
i ⊗F ΓY
(cid:16)ΓX
i ⊗F ΓY

j

(cid:17) = ψk

(cid:16)ΓX
p ⊗F ΓY
(cid:17) ∈ Jk.
p ⊗F ΓY

q

q

j − ΓX

(cid:17), then

p × ΓY

q ⊂ Xp × Yq are path-connected components,

p × ΓY
Indeed, let γ : [0, 1] −→ (X ⊗ Y)k be a path with γ(0) ∈ ΓX
q .
Since each Xk−‘ × Y‘ is open in (X ⊗ Y)k, then the Lebesgue’s number lemma yields a
partition 0 = r0 < r1 < ··· < rN = 1 such that γ ([rn, rn+1]) ⊂ Xin × Yjn and in + jn = k
for all 0 ≤ n < N. Let (xn, yn) = γ (rn), and let ΓX
jn be its path-connected component
in Xin × Yjn. Since (i, j) = (i0, j0) and (p, q) = (iN , jN), then it is enough to show that

j and γ(1) ∈ ΓX

in ×ΓY

i × ΓY

(cid:16)ΓX
in ⊗F ΓY

(cid:17) ∈ Jk

jn − ΓX

in+1 ⊗F ΓY

jn+1

for all

n = 0, . . . , N − 1.

Fix 0 ≤ n < N, assume that in ≤ in+1 , and let ‘ = in+1 − in = jn − jn+1 (the argument is
similar if in ≥ in+1). Since the projection πX (γ ([rn, rn+1])) ⊂ Xin is a path in Xin from xn
in+1 in H0(Xin+1). Similarly, the projection πY (γ ([rn, rn+1])) ⊂
to xn+1, then t‘ΓX
Yjn is a path in Yjn from yn to yn+1, and since jn+1 ≤ jn, then t‘ΓY
jn in H0(Yjn).
This calculation shows that

jn+1 = ΓY

in = ΓX

jn − ΓX
i ⊗F ΓY

(cid:16)ΓX
in ⊗F ΓY
and therefore (cid:16)ΓX
Let us now show that ker(ψk) ⊂ Jk. To this end, let c ∈ ker(ψk) and write

(cid:17) =(cid:16)ΓX
jn+1 − t‘ΓX
(cid:17) ∈ Jk. (end of Claim 1)
jn+1
p ⊗F ΓY
cn ·(cid:16)ΓX
NX

in+1 ⊗F ΓY
j − ΓX

in ⊗F t‘ΓY

in ⊗F ΓY

jn+1

c =

(cid:17)

q

in ⊗F ΓY

jn

(cid:17) ∈ Jk

n=1

where cn ∈ F, and where the indices (in, jn) have been ordered such that there are integers
0 = n0 < n1 < ··· < nD = N, and distinct path-connected components Γ1, . . . , ΓD of
(X ⊗ Y)k so that if nd−1 < n ≤ nd, then ψk
the Γd’s are linearly independent in H0((X ⊗ Y)k), then ψk(c) = 0 implies

(cid:17) = Γd for all d = 1, . . . , D. Since

(cid:16)ΓX
in ⊗F ΓY

jn

nd−1X

n=1+nd−1

cnd =

−cn

, d = 1, . . . , D

75

and therefore

DX

nd−1X

d=1

n=1+nd−1

cn ·

c =

(cid:18)

in ⊗F ΓY
ΓX

jn − ΓX
ind

(cid:19)

⊗F ΓY
jnd

where each summand is an element of Jk, by Claim 1 above. Thus, ker(ψk) = Jk, ψk induces
a natural F-isomorphism

Ψk :(cid:16)

P H0(X ) ⊗F[t] P H0(Y)(cid:17)

−→ H0((X ⊗ Y)k)

k

and letting Ψ = Ψ0 ⊕ Ψ1 ⊕ ··· completes the proof.

We now move onto the main results of this section.

Lemma 3.3.10 (Persistent Eilenberg-Zilber). For objects X ,Y ∈ FTop0 and coefficients
in a field F, there is a natural graded chain equivalence

ζ : P S∗(X ) ⊗F[t] P S∗(Y) −→ P S∗(X ⊗ Y)

with H0(ζ) = Ψ (Lemma 3.3.9), and unique up to a graded chain homotopy.

Proof. The proof follows exactly that of Theorem 2.2.4 [32], using acyclic models [30] (see
also the proof of Theorem 5.3 in [71] for a more direct comparison). Indeed, the functors
in question are acyclic (Lemma 3.3.7), free (Lemma 3.3.8), and naturally equivalent at the
level of zero-th homology (Lemma 3.3.9).

Theorem 3.3.11. If X ,Y ∈ TopN, then there is a natural F[t]-isomorphism

(cid:16)
P S∗(T (X )) ⊗F[t] P S∗(T (Y))(cid:17) ∼= P Hn(X ⊗g Y).

Hn

Proof. Lemma 3.1.15 implies that P Hn(X ⊗g Y) ∼= P Hn(T (X ) ⊗ T (Y)), and Proposition
3.1.17 shows that if 0 < 
 < 1, then

P Hn(T (X ) ⊗ T (Y)) ∼= P Hn(T 
(X ) ⊗ T 
(Y)).

Moreover, since T 
(X ) ∈ FTop0, then the inclusion

P S∗(T 
(X ) ⊗ T 
(Y)) ,→ P S∗(T 
(X ) ⊗ T 
(Y))

76

is a graded chain homotopy equivalence (see [39, Proposition 2.21]), and thus Lemma 3.3.10
shows that

P Hn(X ⊗g Y) ∼= Hn

(cid:16)

P S∗(T 
(X )) ⊗F[t] P S∗(T 
(Y))(cid:17)

.

The 
 can be removed since T 

which completes the proof.

k (X ) deformation retracts onto Tk(X ) (Proposition 3.1.17),

Finally, we obtain

Theorem 3.3.12. Let X ,Y ∈ TopN. Then there is a natural short exact sequence of graded
F[t]-modules

0 → M

(cid:16)

i+j=n

P Hi(X ; F) ⊗F[t] P Hj(Y; F)(cid:17) → P Hn

(cid:16)X ⊗g Y(cid:17) →
P Hi(X ; F), P Hj−1(Y; F)(cid:17) → 0

M

TorF[t]

(cid:16)

which splits, though not naturally.

i+j=n

Remark 3.3.13. If S is either Top, Met, oSimp, or Simp, then following Remark 3.2.1
yields a similar Persistent Künneth theorem in each case.

3.3.4 Barcode Formulae

We now give the barcode formula for the generalized tensor product of objects in SN. Recall
from Theorem 2.3.10 that pointwise finite objects in ModNF can be uniquely decomposed
into interval diagrams. In Proposition 3.3.3, the tensor product and Tor of the associated
interval modules was computed, and the following is a consequence of Theorem 3.3.12.

Theorem 3.3.14. Let X ,Y ∈ SN. Assume that for 0 ≤ i, j ≤ n, the diagrams Hi(X ) and

77

Hj(Y) are pointwise finite. Then Hn(X ⊗g Y) is pointwise finite with barcode

(

bcdn(X ⊗g Y) = [
[

i+j=n

i+j=n

(‘J + I) ∩ (‘I + J)

(
(ρJ + I) ∩ (ρI + J)

)
(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ), J ∈ bcdj(Y)
[
)
(cid:12)(cid:12)(cid:12)(cid:12) I ∈ bcdi(X ), J ∈ bcdj−1(Y)

.

In the above formula, the first union of intervals is associated with the tensor part of
the sequence in theorem 3.3.12, while the second union is associated with the Tor part. The
results are easy to generalize to 0-th persistent homology in the case of finite tensor product
of objects in Top.
Corollary 3.3.15 (0-th persistent homology). Let X1, . . . ,Xp∈ SN, and assume that H0 (Xi)
is pointwise finite for all i ∈ N0. Let bcd0(Xi) be the 0-dimensional barcode of Xi, then
)
, Ii ∈ bcd0(Xi)

(cid:16)X1 ⊗g ··· ⊗g Xp

(cid:17) + Ii

( p\

(cid:17) =

pX

(cid:16)(cid:16)

bcd0

.

‘ − ‘Ii

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ‘ =

‘Ii

i=1

In other words, a bar in X1 ⊗g ··· ⊗g Xp corresponding to p bars, one from each bcd0(Xi),
starts at the sum of the starting points and lives as long as the shortest one.

i=1

Topological inference deals with estimating the homology of a metric space X from a finite
point cloud X, as discussed in Example 2.3.2. Usually, longer bars in a barcode represent
significant topological features. The following corollary allows us to count the number of
bars longer than a threshold 
 > 0 in the barcode of the tensor product.
Corollary 3.3.16 (On high persistence). For n ≥ 0 and 
 > 0, let cX
n (β ≥ 
)
denote the number of bars with length exactly β and at least 
 respectively in bcdn(X ). Assume
that the lengths of longest bars in bcdn(X ) and bcdn(Y) are βX
n respectively, then
(cid:26)
l−1}

n and βY
k , βY
min{βX

n (β) and cX

min{βX

X⊗gY
n

(cid:27))

= max

(

(cid:26)

(cid:27)

β

max
k+l=n

and

cX⊗gY

n

(β ≥ 
) = X

k+l=n

cY
l (β ≥ 
)cX

cY
l (β ≥ 
)cX

k (β ≥ 
).

(3.10)

, max
k+l=n

k , βY
l }
k (β ≥ 
) + X

k+l=n−1

78

X⊗gY
By replacing 
 with β
n

, the formula in equation 3.10 can be used to compute the

number of bars with the longest length in bcdn

(cid:16)X ⊗g Y(cid:17).

3.4 Application: Persistent Homology of N-torus

The Rips persistence diagrams of a standard N-torus equipped with the maximum metric
can be computed from the following two propositions 3.4.1 and 3.4.2. The first is a result
by Adams and Adamaszek [1] that theoretically computes the homotopy type of the Rips
complex of a circle S1, at each scale 
. In the original result, the circle was a equipped with
the geodesic metric but we present a remetrized version with the Euclidean metric:

Proposition 3.4.1 ([1]). The Vietoris-Rips complex R
(S1
with Euclidean metric) is homotopy equivalent to S2l+1 if
 

!

 

2r sin

l

π

2l + 1

< 
 ≤ 2r sin

l + 1
2l + 3

π

!

.

r) of a circle of radius r (equipped

Recall the homology of an m-sphere, with coefficients in some field F, is F in dimensions
i = 0, m and {0} otherwise. Now, Proposition 3.4.1 implies that for a circle of radius r, the
Rips persistence diagrams in odd dimensions are singleton as multisets and empty in positive
even dimensions:

(cid:17)

, 2r sin(cid:16)

π l+1
2l+3

(cid:17)(cid:17)o

π l

2l+1

(cid:16)

S1
r

(cid:17) =

bcdR

i



nh2r sin(cid:16)

{[0,∞)}
∅

if i = 2l + 1

if i = 0

if i = 2l + 2

(3.11)

for l ∈ N0. The second proposition is an extension of Corollary 3.2.6 to an N-fold product
of metric spaces:
Proposition 3.4.2. Let (X1, d1), . . . , (XN , dN) be finite metric spaces and let bcdR
be the n-th dimensional persistence diagram of the Rips filtration R(X, dX) := {R
(X, dX)}
≥0.

n (X, dX)

79

Then,

bcdR

n (X1 × ··· × XN , dmax) =

(cid:26) N\

m=1

Im

(cid:12)(cid:12)(cid:12)(cid:12) Im ∈ bcdR

im(Xm, dm)

(cid:27)

[

NP

m=1

im=n

for all n ∈ N0, where dmax is the maximum metric.

Consider the Vietoris-Rips complexes on the N-torus TN := S1

equipped
with the maximum metric. Then by Propositions 3.4.1 and 3.4.2, we obtain the following
theorem about the Rips persistence of a N-torus:

r1 × ··· × S1
rN

Theorem 3.4.3. Let TN := S1
be the N-torus equipped with the maximum
metric dmax. Then for each homological dimension p, the Rips persistence barcode of TN
can be computed as:

r1 × ··· × S1
rN

(cid:16)TN , dmax

bcdR

p

(cid:17) = [
NP

m=1

im=p

 N\

m=1

(cid:12)(cid:12)(cid:12)(cid:12) Im ∈ bcdR

im

(cid:16)

S1
rm

Im

(cid:17)  .

More explicitly,

√

o

bcdR
1

in dimension 1, and

3rn] (cid:12)(cid:12)(cid:12)n = 1, . . . , N

(cid:16)TN , dmax
(cid:17) =n[0,
3 min{rn, rm}i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n, m = 1, . . . , N
(h0,
in dimension 2. Let r = min{r1, . . . , rN}. Then,
(cid:17) bars in bcdR
Corollary 3.4.4. There are exactly (cid:16)N

(cid:16)TN , dmax

(cid:16)TN , dmax

(cid:17) =

bcdR
2

√

p

p

)

(cid:17) that are born at 0 and

live longer than

or of length atleast

√
3r.

√

This is true because all the bars used in the persistent Künneth formula are either infinite,

3r, or shorter, depending on the dimension.

80

CHAPTER 4

QUASIPERIODICITY

In this chapter, we develop the general theory of sliding window embeddings of quasiperi-
odic functions and their persistent homology. We define quasiperiodic functions in a rigorous
manner and study their properties in section 4.1 and write a Fourier-like decomposition in
section 4.2. In section 4.3, we show that the sliding window embeddings of quasiperiodic
functions, under appropriate choices of the embedding dimension d and time delay τ, are
dense in higher dimensional tori. We also explicitly provide methods to choose these parame-
ters. In section 4.4, we prove lower bounds on Rips persistent homology of these embeddings.
The last section of this chapter is dedicated to computational results and an application to
music. Parts of this chapter will appear in [36].

4.0.1 Notation

Let T = R/2πZ ’ S1, that is, [0, 2π] with the endpoints identified. Similarly for N ∈ N,
let TN = (R/2πZ)N ’ S1 × ··· × S1. Let v = (v1,··· , vM) ∈ RM such that none of the
coordinates vm are 0, then let Tv define

, 2π

!

v1

,  2π

!

Z

.

vM

Tv = R

Z

× ··· × R

c denotes the circle of radius c, then for c = (c1, . . . , cN) ∈ CN where none of the cn are

If S1
0, let TNc denote

TNc := S1|c1| × ··· × S1|cN|.

4.1 Quasiperiodic functions

In this section, we provide a general definition of a quasiperiodic function for a given

incommensurate set of harmonics and give some examples.

81

Definition 4.1.1. Let f : R → C be a function and let ω = {ω1, . . . , ωN} ⊂ R \ {0} be an
incommensurate set, that is, ω is linearly independent over Q. Then we call f a quasiperiodic
function with the set of harmonics ω if there exists a function F : RN → C

(t1, , . . . , tN) → F(t1, , . . . , tN)

such that

1. F(t, . . . , t) = f(t) for all t ∈ R,

2. for all 1 ≤ n ≤ N, and ∀(cid:16)

(cid:17) ∈ RN−1, the one variable function
n+1, . . . , t∗

N))

t∗
1, . . . , t∗
tn → F((t∗

n−1, t∗
1, . . . , t∗

n+1, . . . , t∗
n−1, tn, t∗

N

is periodic with time period 2π
ωn

, and

3. N is the lowest such integer.

We will call F a parent function of f.

The condition that N must be minimal is necessary. In its absence, f(t) = eiteiπt will
have two parent functions: F(t1, t2) = eit1eiπt2 and G(t1) = ei(π+1)t1. The function f is
clearly periodic with a single frequency π + 1.

(cid:27)

(cid:26)2π

Remark 4.1.2. Note that every periodic function can be viewed as a quasiperiodic function.
Indeed, if f is periodic with time period tf, then it is quasiperiodic with the set of harmonics
ω =
and the associated parent function F(t1) := f(t1). However, the incommensurate-
ness of the harmonics implies that not all quasiperiodic functions are periodic. For example,
f(t) = eit + eiπt is quasiperiodic with {1, π} and parent function F(t1, t2) = eit1 + eiπt2, but

(cid:17) of the time periods of individual terms of f does not

it is not periodic because lcm(cid:16)2π1 , 2π

tf

π

exist.

The following example demonstrates that for a quasiperiodic function f with a set of

harmonics ω, its parent function may not be unique.

82

Example 4.1.3. The function f(t) = eit + eiπt is quasiperiodic with the set of harmonics
{1, π}. Let F, G : R2 → C be defined as

F(t1, t2) = eit1 + eiπt2
G(t1, t2) = eit2 + eiπt1.

Clearly, f is the restriction to the diagonal of the functions F and G. Moreover, F(t1, t∗
2)
and G(t1, t∗
2) are periodic in t1 with periods 2π and 2 respectively for all choices of t?2, and
F(t∗
1, t2) are periodic in t2 with periods 2 and 2π respectively for all choices of
t∗
1. Therefore, F and G are distinct parent functions of f.

1, t2) and G(t∗

In the next proposition, we show that for a continuous quasiperiodic f associated to a

given set ω, its continuous parent function is unique up to a permutation of variables:

Proposition 4.1.4. If f is a continuous quasiperiodic function with a set ω = {ω1, . . . , ωN},
then its continuous parent function is unique up to a permutation of variables.

(cid:26)(cid:18)

k1 2π

ω1 + t, . . . , kN

2π
ωN

+ t

(cid:19)(cid:12)(cid:12)(cid:12)t ∈ R
(cid:27)

Proof. Let f : R → C be a quasiperiodic function with the incommensurate set {ω1, . . . , ωN}
and let there be two continuous parent functions F, G : RN → C. After a permutation of
variables, assume that the functions F and G are periodic with the same time period 2π
in tn.
ωn
By definition, F and G agree on the diagonal 40 = {(t, . . . , t)|t ∈ R}. Therefore, they agree
on the lines 4k =
for each k = (k1, . . . , kN) ∈ ZN. We
claim that the set of all 4k is dense in the set of all lines parallel to the diagonal. For
an arbitrary (t1, . . . , tN) ∈ RN, consider the hyperplane V : t1 = t1. We will show that
there is a line 4k intersecting V at a point arbitrarily close to (t1, . . . , tN). Any line 4k
intersects V at a point
which yields that t1 = t1 and for
ω2 + t, . . . , kN
2 ≤ n ≤ N, tn = kn
2π
ω1 . Without loss of generality, we can assume that t1 = 0.
ωn
We want to show that the set of the intersection points I is dense in V . We can rewrite
tn = 2π
ωn
V . Then the intersection points satisfy s1 = t1 = t1 and sn = kn − k1 ωn

(cid:17). Define sn = ωn2π tn for 2 ≤ n ≤ N. Call this rescaled hyperplane
ω1 for 2 ≤ n ≤ N.

(cid:18)
t1, k2 2π
+ t1 − k1 2π

kn − k1 ωn
ω1

(cid:16)

(cid:19)

2π
ωN

+ t

83

ωN
ω1

ω1 , . . . ,

o is incommensurate. Then by Kronecker’s theorem (theorem 2.5.2),
Clearly, n ω2
for any arbitrary (t1, s2, . . . , sN) ∈ V , we can find an intersection point arbitrarily close to
it. Therefore, we can find an intersection point arbitrarily close to (t1, . . . , tN). This proves
our claim. Therefore, F and G agree on a dense subset of the set of lines parallel to the
diagonal. By their continuity, they agree everywhere.

Before we move on to writing a Fourier like series for a quasiperiodic function f, we want

to mention that some functions can be quasiperiodic with different sets of harmonics:

Example 4.1.5. Observe that the function f(t) = (cid:16)

eit + 1(cid:17)

eiπt is quasiperiodic with sets
{1, π} and {1, π+1}: it is the restriction to the diagonal of two parent functions F, G : R2 →
C defined as

F(t1, t2) =(cid:16)

eit1 + 1(cid:17)

eiπt2
G(t1, t2) = ei(π+1)t1 + eiπt2.

2) and G(t1, t∗

F(t1, t∗
of t?2, and F(t∗

2) are periodic in t1 with periods 2π and 2π

π+1 respectively for all choices
1, t2) are both periodic in t2 with periods 2 for all choices of t∗
1.

1, t2) and G(t∗

4.2 “Fourier series” of a quasiperiodic function

In this section, we adapt the theory of multivariate Fourier analysis to write a Fourier-like

series for the function f. For a more comprehensive review, see [38, Chapter 3].

4.2.1 The iterated Fourier series
Let f : R → C be a quasiperiodic function with the incommensurate set {ω1, . . . , ωN} and
parent function F : RN → C. Let ω denote the frequency vector (ω1, . . . , ωN) and recall the
notation (section 4.0.1) for Tω:

, 2π

!

Z

ω1

Tω = R

,  2π

!

Z

.

ωN

× ··· × R

84

Furthermore assume that the parent function F belongs to C1(Tω). Recall that by definition
of a quasiperiodic function, each marginal

(cid:16)

Fn(tn) := F

t∗
1, . . . , t∗

n−1, tn, t∗

n+1, . . . , t∗

N

(cid:17)

is a one variable periodic function in tn with the period 2π
ωn

.

Lemma 4.2.1. If g : X → C is a continuously differentiable function on a finite measure
space X ⊂ Rm, then it is square integrable on X.

The next proposition shows that for each such F, all its marginal functions Fn are

continuously differentiable.

Proposition 4.2.2. Let TωN = R

(cid:30)(cid:18) 2π

ωN

(cid:19)

Z

. If F ∈ C1 (Tω), then FN ∈ C1 (TωN ).

Proof. Since F is continuously differentiable on Tω, all its partial derivatives exist and are
continuous, which yields that FN is continuously differentiable with respect to tN. Therefore,
FN ∈ C1 (TωN ).
For each choice of (cid:16)
t∗
1, . . . , t∗

(cid:17) ∈ Tω1 × ··· × TωN−1, we obtain a Fourier series
(cid:17) = X

t∗
1, . . . , t∗

t∗
1, . . . , t∗

eikN ωN tN

N−1, kN

(cid:16)

ˆF

(cid:16)

F

N−1, tN

N−1

(cid:17)

kN∈Z

where

(cid:16)

ˆF

t∗
1, . . . , t∗

N−1, kN

(cid:17) = ωN

2π

(cid:16)

F

2π

ωNZ

0

t∗
1, . . . , t∗

N−1, tN

(cid:17)

e−ikN ωN tN dtN

are its Fourier coefficients. By proposition 2.5.1, the series converges pointwise to FN. The
next proposition shows that the coefficient functions ˆF (t1, . . . , tN−1, kN) for each choice of
kN ∈ Z are square integrable on Tω1 × ··· × TωN−1.
Proposition 4.2.3. If F ∈ C1 (Tω), then for all kN ∈ Z, the N − 1 variable coefficient

functions ˆF(t1, . . . , tN−1, kN) ∈ L2(cid:16)Tω1 × ··· × TωN−1(cid:17).

85

Proof. The proof follows directly from the definition:

Z

Z

Tω1×···×TωN−1

=

Tω1×···×TωN−1
Z

≤ ωN
2π

(cid:12)(cid:12)(cid:12) ˆF(t1, . . . , tN−1, kN)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

ωNZ

ωN
2π

2π

0

2π

ωNZ

dtN−1 ··· dt1

F(t1, . . . , tN)e−ikN ωN tN dtN

2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

dtN−1 ··· dt1

|F(t1, . . . , tN)|2 dtN dtN−1 ··· dt1 < ∞

Tω1×···×TωN−1

0

where the last term is finite because F ∈ C1(Tω) and Lemma 4.2.1.

Now observe that for each (t∗
one variable function defined by

1, . . . , t∗

N−2) ∈ Tω1 × ··· × TωN−2 and each kN ∈ Z, the

ˆF(t∗

1, . . . , t∗

N−2.tN−1, kN) = ωN
2π

2π

ωNZ

0

F(t∗

1, . . . , t∗

N−2, tN−1, tN)e−ikN ωN tN dtN

is periodic in tN−1 with period
Proposition 4.2.4. For each (t∗
function ˆF(t∗

1, . . . , t∗

2π
.
ωN−1
1, . . . , t∗

N−2, tN−1, kN) ∈ L2(cid:16)TωN−1(cid:17) ∩ C1(cid:16)TωN−1(cid:17).

N−2) ∈ Tω1 × ··· × TωN−2 and each kN ∈ Z, the

Proof. Similar to the proof of proposition 4.2.2.

N−2, tN−1, kN) as a Fourier series which converges

ˆF(t?1, . . . , t?

N−2, kN−1, kN)eikN−1ωN−1tN−1

Therefore, we can write ˆF(t∗

1, . . . , t∗
N−2, tN−1, kN) = X

kN−1∈Z

pointwise to it:

ˆF(t?1, . . . , t?

which is now equal the following double sum:

X
kN∈Z

X
kN−1∈Z

ˆF(t?1, . . . , t?

N−2, kN−1, kN)eikN−1ωN−1tN−1eikN ωN tN ,

86

where ˆF(t∗

ωN−1ωN
(2π)2

1, . . . , t∗
ωN−1Z
2π

N−2.kN−1, kN) is given by
ωNZ

e−ikN−1ωN−1tN−1

2π

0

0

F(t∗

1, . . . , t∗

N−2, tN−1, tN)e−ikN ωN tN dtN dtN−1.

Similar to Proposition 4.2.3, we can show that if

ˆF(t1, . . . , tN−1, kN) ∈ L2(Tω1 × ··· × TωN−1),

then ˆF(t1, . . . , tN−2, kN−1, kN) ∈ L2(Tω1 × ··· × TωN−2) for all pairs (kN−1, kN) ∈ Z2:

dtN−2 ··· dt1

Z

Z

(cid:12)(cid:12)(cid:12) ˆF(t1, . . . , tN−2, kN−1, kN)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

ωN−1Z

ωN−1
2π

2π

0

2π

ωN−1Z
(cid:12)(cid:12)(cid:12) ˆF(t1, . . . , tN−1, kN)(cid:12)(cid:12)(cid:12)2

0

Tω1×···×TωN−2

=
Tω1×···×TωN−2
Z
Z

≤ ωN−1
2π
≤ ωN−1
2π

Tω1×···×TωN−2

Tω1×···×TωN−1

ˆF(t1, . . . , tN−1, kN)e−ikN−1ωN−1tN−1dtN−1
(cid:12)(cid:12)(cid:12) ˆF(t1, . . . , tN−1, kN)(cid:12)(cid:12)(cid:12)2

dtN−1 ··· dt1

dtN−1 ··· dt1 < ∞.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

2
dtN−2 ··· dt1

Repeating the above arguments, we obtain that for all (kn+1, . . . , kN) ∈ ZN−n, the inter-
mediate ˆF(t1, . . . , tn, kn+1, . . . , kN) will be in L2(Tω1 × ··· × Tωn) if F ∈ L2(Tω). Using
induction, we obtain

F(t1, . . . , tN) = X

··· X

k1∈Z

kN∈Z

ˆF(k1, . . . , kN)ei(k1ω1t1+···+kN ωN tN )

(4.1)

where the convergence of series to F is pointwise because all intermediate ˆF’s are continu-
ously differentiable. Note that for k = (k1, . . . , kN) ∈ ZN, the coefficients ˆF(k) are given
by:

ˆF(k) =

e−ik1ω1t1

F(t1, . . . , tN)e−ikN ωN tN dtN . . . dt1.

NQ

ωn
n=1
(2π)N

Z 2π

ω1

0

Z 2π

ω2

0

Z 2π

ωN

0

···

87

To be able to rearrange the terms of the series in equation 4.1, we need it to be absolutely
convergent. To prove this, we investigate the coefficients ˆF(k) in the next section. We
finish this section with a remark about the periodicity of the partial derivatives of the parent
function which will be useful in proof of Theorem 4.2.6.
Remark 4.2.5. Let f be a quasiperiodic function and let F : RN → C be its parent function.
Suppose for some 1 ≤ n ≤ N and r ∈ N, the r-th partial derivative ∂r
will be periodic in tn with the same period 2π
ωn
it will be periodic in tm with 2π
ωm

nF exists, then ∂r
nF
nF is dependent on tm, then

. If for m 6= n, ∂r

.

Z = nk = (k1, . . . , kN) ∈ ZN(cid:12)(cid:12)(cid:12) ∀n , |kn| ≤ Z

4.2.2 Analyzing the coefficients ˆF(k)
o denote the box of lengths
For Z ∈ N0, let IN
2Z centered around (0, . . . , 0) ∈ ZN. Moreover, let k (cid:12) ω = (k1ω1, . . . , kN ωN) denote the
Hadamard product of two vectors k and ω. The next theorem provides bounds on each
coefficient ˆF(k) in terms of its corresponding coefficient in the series decomposition of a
certain partial derivative. Furthermore, it concludes that the sequence of coefficients is
absolutely convergent.

Theorem 4.2.6. Let r ∈ N and k, l ∈ ZN. Suppose that ∂lF exist and are integrable for all
||l||1 ≤ r. Then

√
N
||k (cid:12) ω||r2
where n satisfies |knωn| = ||k (cid:12) ω||∞ and d∂r

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) ≤

r

nF(k)(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) d∂r

nF is the r-th (partial) derivative of F with

respect to tn. As a consequence, the sequence of coefficients is absolutely convergent:

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) < ∞.

X

k∈ZN

88

Proof. Fix k ∈ ZN \ {0} and let n be the coordinate such that |knωn |= ||k (cid:12) ω||∞. Then
kn 6= 0. Recall that
NQ

2π

2π

ω1Z

ω2Z

2π

ωNZ

F(t1, . . . , tN)e−ikN ωN tN dtN . . . dt1.

ˆF(k) =

ωn
n=1
(2π)N

e−ik1ω1t1

···

0

0

0

Using Fubini’s theorem, we can change the order so that the integral with respect to tn is
the inner most and looks like

2π

ωnZ

0

F(t1, . . . , tN)e−iknωntndtn

Integrating by parts with respect to tn yields

2π

ωnZ

0

F e−iknωntndtn =

" e−iknωntn
−iknωn

# 2π

ωn

0

F

−

2π

ωnZ

0

e−iknωntn
−iknωn

∂nF dtn

where the first term on the right hand side vanishes. The second term can be further
integrated using by parts to obtain

and the first term vanishes again. After a total of r such iterations, we obtain

" e−iknωntn
(−iknωn)2 ∂nF

# 2π

ωn

0

+

e−iknωntn
(−iknωn)2 ∂2

nF dtn

F e−iknωntndtn = (−1)r

e−iknωntn
(−iknωn)r ∂r

nF dtn

−

2π

ωnZ

0

2π

ωnZ

0

2π

ωnZ

0

2π

ωNZ

0

2π

ω2Z

0

···

which gives us upon rearrangement

1

NQ
(iknωn)r d∂r

ωn
n=1
(2π)N
1

(iknωn)r

nF(k)

2π

ω1Z

0

ˆF(k) =

=

e−ik1ω1t1

nF(t1, . . . , tN)e−ikN ωN tN dtN . . . dt1
∂r

where the last step results from remark 4.2.5. Using the fact that ||k(cid:12)ω||2 ≤ √
√
N | knωn |, we obtain

N||k(cid:12)ω||∞ =

(4.2)

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) ≤

r

√
||k (cid:12) ω||r2

N

(cid:12)(cid:12)(cid:12) d∂r
nF(k)(cid:12)(cid:12)(cid:12) ,

89

which proves the first part of the theorem. Now, we will show that

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) < ∞,

X

k∈ZN

By equation 4.2, we obtain

X

k /∈IN
Z

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) ≤ X

k /∈IN
Z

r

√
||k (cid:12) ω||r2

N

| d∂r
nF(k)|

Note that n depends on k and can be written as n(k). Using Cauchy-Schwarz theorem and
Parseval’s identity, the right hand term can be bounded:

X

k /∈IN
Z

r

√
|k (cid:12) ω|r| (cid:92)∂r
N

n(k)F(k)| ≤

√

r

N

 X

k /∈IN
Z



1/2 X

k /∈IN
Z

1

||k (cid:12) ω||2r2



1/2

.

| (cid:92)∂r
n(k)F(k)|2

nF is continuous, so it is square integrable and the coefficients are

However, for each n, ∂r
square summable:

k∈ZN
Hence, we can rearrange terms and get



1/2

≤

| (cid:92)∂r
n(k)F(k)|2

 X

k /∈IN
Z

X
 X

k /∈IN
Z

| d∂r
nF(k)|2 < ∞.

(4.3)

NX

n=1

| d∂r
nF(k)|2

1/2



 NX

n=1

=

X

k /∈IN
Z

1/2



| d∂r
nF(k)|2

where the last term is finite because of equation 4.3. Before proceeding further, we denote
by J N

Z , the box

by BN

Z , the ball

around 0 = (0, . . . , 0) ∈ RN, and by EN

,

nx = (x1, . . . , xN) ∈ RN (cid:12)(cid:12)(cid:12) ||x||∞ ≤ Z
o
nx = (x1, . . . , xN) ∈ RN (cid:12)(cid:12)(cid:12) ||x||2 ≤ Z
o
!1/2

Z , the ellipsoid-ball
+ ··· + y2
N
ω2
N

y = (y1, . . . , yN) ∈ RN

  y2

1
ω2
1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

,



≤ Z

90

that we obtain upon a change of variables ωnxn = yn for each n. Observe that
1/2

 X

k /∈IN
Z

1/2



1

||k (cid:12) ω||2r2

(cid:17)r dx1dx2 . . . dxN

(cid:17)r dx1dx2 . . . dxN



(cid:17)r dy1dy2 . . . dyN

1/2

1/2



.

1 + ··· + x2

N ω2

N

(cid:16)

x2
1ω2
(cid:16)

1

1

N

1 + ··· + x2
N ω2
1ω2
x2
(cid:16)
!1/2



Z

y /∈EN
Z

1
|ωn|

1

1 + ··· + y2
y2

N

≤

≤

=

x /∈JN
Z

x /∈BN
Z

Z

Z



  NQ


n=1

Z

Now, let Zm := Zωmin = Z min{ω1, . . . , ωN}. Then the above is bounded by
1/2

!1/2

  NQ

n=1

1
|ωn|

y /∈BN
Zm

(cid:16)

1

1 + ··· + y2
y2

N

(cid:17)r dy1dy2 . . . dyN

,



which in higher dimensional spherical coordinates can be written as

Z

σ

∞Z

Zm

1/2

1
(ρ)2r ρdρdσ

  NQ

n=1

!1/2

1
|ωn|

where dσ is the differential surface area of unit sphere SN−1 and ρ is the distance of a point
from the center. Evaluating the integrals yields

1/2
1/2

1

(ρ)2r−1 dρ

Zm

 ∞Z
Area(SN−1)(cid:17)1/2
 ρ2−2r
Area(SN−1)(cid:17)1/2
Area(SN−1)(cid:17)1/2 Z1−r

2 − 2r

(cid:16)

(cid:16)
(cid:16)

m√
2r − 2.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

∞

Zm

n=1

  NQ
  NQ
  NQ

n=1

n=1

!1/2
1
|ωi|
!1/2
1
|ωi|
!1/2
1
|ωi|

=

=

91

Summarizing, we obtain
√

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) ≤

X

k /∈IN
Z

Area(SN−1)(cid:17)1/2
r(cid:16)
!1/2
N
√
ωr−1
2r − 2
min

  NQ

|ωn|

n=1

 NX

n=1

X

k /∈IN
Z

1/2



1
Zr−1

| d∂r
nF(k)|2

(4.4)

where the right hand side is finite if Z > 0 and r ∈ N. Since IN
we obtain that

Z has finitely many terms,

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12) < ∞.

X

k∈ZN

(4.5)

Therefore, due to the absolute convergence of the coefficients, equation 4.1 can be rewrit-

ten as:

F(t1, . . . , tN) = X

k∈ZN

ˆF(k)ei(k1ω1t1+···+kN ωN tN ),

where k = (k1, . . . , kN) ∈ ZN. Along the diagonal, we get an iterated Fourier series for the
quasiperiodic function f:

f(t) = F(t, . . . , t) = X

k∈ZN

it
ˆF(k)e

knωn = X

k∈ZN

ˆF(k)eithk,ωi

(4.6)

NP

n=1

where ω = (ω1, . . . , ωN) ∈ RN and h·,·i is the inner product (dot product) on RN. Moreover,
the series in equation 4.6 converges everywhere. Next, we move to sliding window embedding
of the function f.

92

4.2.3 Approximation theorems for sliding window embeddings

From Definition 2.4.4 and Equation 4.6, the sliding window embedding of a quasiperiodic
function f at t with parameters d and τ is:



P
P

k∈ZN

ˆF(k)eihk,ωit
ˆF(k)eihk,ωi(t+τ)

P

...

ˆF(k)eihk,ωi(t+dτ)

k∈ZN



= X

k∈ZN

ˆF(k)eihk,ωituk





f(t)

f(t + τ)

...

f(t + dτ)

SWd,τ f(t) =

=

k∈ZN

where uk is the vector

uk =



1

eihk,ωi(τ)

...

eihk,ωi(dτ)

 .

Remark 4.2.7. We make a few observations.

1. uk = SWd,τ eithk,ωi |t=0.
2. If k = l, then huk, uli = d + 1.
3. hk, ωi = 0 if and only if k1 = ··· = kN = 0
4. hk, ωi = hl, ωi if and only if k = l.

For Z ∈ N0, we define the polynomial SZ f by truncating the series:

SZ f(t) := X

k∈IN
Z

ˆF(k)eihk,ωit,

where IN

Z is the box

Z =nk ∈ ZN (cid:12)(cid:12)(cid:12) ||k||∞ ≤ Z

IN

o

.

93

Then SZ f(t) converges to f(t) for each t ∈ R as Z → ∞. Since SWd,τ is a bounded linear
operator (and hence, continuous) by Proposition 2.4.8, SWd,τ SZ f converges to SWd,τ f
pointwise as well. Let

RZ f(t) := f(t) − SZ f(t) = X

ˆF(k)eihk,ωit

k /∈IN
Z

be the remainder term of polynomial SZ f(t). Then the following result shows that the
bound on the sliding window embedding of the remainder term RZ f is controlled by the
regularity of the function f. In particular, the more differentiable f is, the faster is the rate
of convergence.

Theorem 4.2.8. Let r ∈ N \ {1}. If F ∈ Cr(cid:16)RN , C(cid:17), then for all t ∈ R
1/2 √

||RZ(∂r

nF)||2

≤

d + 1
Zr−1 .


1/2

X

k /∈IN
Z

1/2

| d∂r
nF(k)|2

1
Zr−1

||RZ(∂r

nF)||2
2

1
Zr−1

 NX

n=1

1/2 √

d + 1
Zr−1 .

||RZ(∂r

nF)||2
2

Proof. We begin with the remainder term RZ f:

|RZ f(t)| =

n=1

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

n=1

|ωn|

X

  NQ

r(cid:16)
√
N
√
ωr−1
2r − 2
min

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)SWd,τ f(t) − SWd,τ SZ f(t)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
Area(SN−1)(cid:17)1/2
 NX
!1/2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X
ˆF(k)eihk,ωit
Area(SN−1)(cid:17)1/2
r(cid:16)
!1/2
N
√
ωr−1
2r − 2
Area(SN−1)(cid:17)1/2
r(cid:16)
min
√
!1/2
N
√
ωr−1
2r − 2
min

k /∈IN
Z
√

k /∈IN
Z

|ωn|

|ωn|

n=1

=

≤

  NQ
  NQ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)SWd,τ f(t) − SWd,τ SZ f(t)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
r(cid:16)
√
≤√
N
√
ωr−1
2r − 2
min

d + 1||RZ f||∞ ≤

n=1

Area(SN−1)(cid:17)1/2
!1/2

  NQ

|ωn|

n=1

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12)
 NX
 NX

n=1

n=1

94

where the second inequality comes from Equation 4.4. Therefore,

Corollary 4.2.9. If X = SWd,τ f(T) and Y = SWd,τ SZ f(T) for T ⊂ R, then

dB

(cid:16)bcdR
Area(SN−1)(cid:17)1/2
!1/2

  NQ

n=1

|ωn|

√

r(cid:16)
N
√
ωr−1
2r − 2
min

for

ξ =

i (X), bcdR

i (Y )(cid:17) ≤ 2ξ
 NX

||RZ(∂r

n=1

1/2 √

d + 1
Zr−1

nF)||2
2

Proof. Let 
 > ξ, then by theorem 4.2.8, Y ⊂ X
 and X ⊂ Y 
 and therefore dH(X, Y ) ≤ 
.
Since this is true for every 
 > ξ, this gives us dH(X, Y ) ≤ ξ, which from the stability
theorem [19] gives us

(cid:16)bcdR

dB

i (Y )(cid:17) ≤ 2ξ.

i (X), bcdR

Corollary 4.2.9 implies that the Rips persistent homology barcodes bcdR

i (SWd,τ SZ f)
converge to bcdR
i (SWd,τ f) as Z → ∞. This suggests that one could replace the study of
a quasiperiodic function f with that of its approximation SZ f. Accordingly, we study the
geometric structure of the sliding window embedding of the truncated polynomial SZ f next.
4.3 The geometric structure of SWd,τ SZf

We dedicate this section to showing that under appropriate conditions on d, τ, and Z, the
closure of the sliding window embedding of the truncated polynomial SZ f, is homeomorphic
to the N-dimensional torus. To begin, let α be the cardinality of support of the coefficient
function

ˆF : IN

Z → C
k → ˆF(k)

95

o. Observe that 0 ≤ α ≤ (2Z + 1)N. Choose
Z , say k1,··· , kα. Let xZ,f(t) be a α × 1 vector

where IN
an enumeration of points k ∈ Supp( ˆF) ⊂ IN
given by

Z is the box nk ∈ ZN (cid:12)(cid:12)(cid:12) ||k||∞ ≤ Z


xZ,f(t) =

 .

ˆF(k1)eihk1,ωit

...

ˆF(kα)eihkα,ωit.

Next, we present a corollary to Theorem 2.5.2 that we will utilize to prove the main result
(Theorem 4.3.5) of this section.

Corollary 4.3.1 (a corollary to Kronecker’s theorem). An α-tuple β = (β1, . . . , βα) of real
numbers spans a N-dimensional vector space over Q if and only if the collection {zβ | z ∈ Z}

is dense in an embedding of TN in Tα, which is equivalent to n(eiβ1z,··· , eβαz)(cid:12)(cid:12)(cid:12) z ∈ Zo

being dense in the image of a homeomorphism from TN to Tα.

Proof. Let {β1, . . . , βα} span a N dimensional vector space over Q. Without loss of gen-
erality, let us assume that the β1, . . . , βN form a basis of said vector space. Then for
N + 1 ≤ j ≤ α, βj depends linearly on β1, . . . , βN, i.e. for appropriate coefficients cj,n ∈ Q,
we have

We want to show that {zβ | z ∈ Z} is dense in a space homeomorphic to TN = (R/2πZ)N
in Tα = (R/2πZ)α. It is equivalent to showing that

NX

n=1

βj =

cj,nβn.

n(eiβ1z,··· , eiβαz)(cid:12)(cid:12)(cid:12) z ∈ Zo

is dense in a space homeomorphic to TN in Tα. Observe that

βj =

if and only if

NX

n=1

cj,nβn

eiβj t = (eiβ1t)cj,1 ··· (eiβN t)cj,N .

96

Define a map φ from TN → Tα by

φ(z1, . . . , zN) = (z1, . . . , zN , ˜zN+1, . . . , ˜zα),

(4.7)

cj,1
where ˜zj = z
1

cj,2
z
2

··· z

cj,N
N . Observe that the set

n(eiβ1z,··· , eiβαz)(cid:12)(cid:12)(cid:12) z ∈ Zo ⊂ φ(TN) ⊆ Tα.

By definition, φ is injective, continuous, and it is vacuously surjective onto its own image.
The inverse from the image φ(TN) to TN is just the projection map and that is continuous.
Therefore, φ is a homeomorphism. In particular, it is continuous and surjective (onto φ(TN))
and since the set

is dense in TN by Theorem 2.5.2, its image under φ,

n(eiβ1z,··· , eiβN z)(cid:12)(cid:12)(cid:12) z ∈ Zo
n(eiβ1z,··· , eβαz)(cid:12)(cid:12)(cid:12) z ∈ Zo

is dense in φ(TN). This proves the ‘if’ part of our claim.

For the other direction, let {zβ | z ∈ Z} be dense in φ(TN) ⊂ Tα for some homeo-
morphism φ and let (β1, . . . , βα) span an L-dimensional vector space for L 6= N. From
the first part of the proof, we obtain that {zβ | z ∈ Z} is dense in ϕ(TL) ⊂ Tα for some
homeomorphism ϕ. Therefore,

(cid:16)TL(cid:17) = {zβ | z ∈ Z} = φ(TN)

ϕ

and since homeomorphism is an equivalence relation, we get that TL ’ TN, which is a
contradiction. This finishes the proof.

Remark 4.3.2. From the definition of φ in the proof of Corollary 4.3.1, we observe that for
any two α-tuples β and ˜β spanning a N dimensional vector space over Q, the closure of Zβ
and Z ˜β depends only on how the dependent βj’s (and ˜βj’s) depend on the basis of N linear

97

independent reals, instead of depending the actual values βj’s (respectively ˜βj’s). Assume
that the first N reals in both tuples form linearly independent sets and for N + 1 ≤ j ≤ α,

NX

n=1

NX

n=1

βj =

cj,nβn

and

˜βj =

˜βn

cj,n

for the same dependency coefficients cj,n ∈ Q. Then both Zβ and Z ˜β are dense in φ(TN)
for

cj,1
where ˜zj = z
1

cj,2
z
2

φ(z1, . . . , zN) = (z1, . . . , zN , ˜zN+1, . . . , ˜zα),
cj,N
N .

··· z

For a better intuition of Corollary 4.3.1, see figure 4.1. We begin with four triples of real
3}, the second is a triple {√
3},
2,
3}, and the last is the triple {√
√
3, π}. For each
2,

numbers: the first is equivalent to a pair {√
2,
the third is a triple {√
of these tuples β, we compute Zβ:

√
2 + 2

√
3, 3

√
3,

2+

√

√

√

√

2,

√
Figure 4.1: The sets Zβ for tuples β = (
(second), {√

√
√
√
√
3) ≡ (
2,
2,
3, 0) (left), β = (
√
√
3} (third), and β = (
3, π) (right).
2,

√
3, 3

2 + 2

√

√

2,

2,

√

√
3,

2+

√

3)

The next lemma shows that if Z ∈ N0 is large enough such that Supp( ˆF) contains N
linearly independent vectors, then the set XZ,f = {xZ,f(s) | s ∈ Z} is dense in an embedding
of TN in Tα. For general Z, if M ≤ N is the largest cardinality of a linearly independent
set of vectors in Supp( ˆF), then XZ,f is dense an in embedding of TM in Tα.
Lemma 4.3.3. If Z ∈ N0 is large enough such that Supp( ˆF) contains N linearly independent
vectors, then the set XZ,f is dense in a space homeomorphic to a torus TN embedded in

= ×

k∈Supp( ˆF)

TαˆF

S1
| ˆF(k)|.

98

Proof. Recall that the exponential functions eiat and eibt are equal if and only if a = b.
Moreover, recall from Remark 4.2.7 that the inner product values hk, ωi are different for
distinct vectors k. By the condition on Z, there exist N linearly independent vectors in
Supp( ˆF), say k1, . . . , kN. Then for all j ∈ Supp( ˆF),

for cj,n ∈ Q. Observe that the linearity of inner products implies that for all kj ∈ Supp( ˆF),

kj :=

cj,nkn

NX

n=1

NX

n=1

hkj, ωi =

cj,nhkn, ωi.

Therefore the tuple β = (hk1, ωi, . . . ,hkα, ωi) spans an N-dimensional vector space over Q.

From Corollary 4.3.1, we get that the collection n(eiβ1z,··· , eβαz)(cid:12)(cid:12)(cid:12) z ∈ Zo is dense in an

embedding of TN in Tα. Since the multilinear scaling given by

Tα → ×

k∈Supp( ˆF)

S1
| ˆF(k)|

(z1,··· , zα) → ( ˆF(k1)z1,··· , ˆF(kα)zα)

is a homeomorphism, it follows that XZ,f is dense in an embedding of TN in ×

k∈Supp( ˆF)

Now, let ΩZ,f be the (d + 1) × α matrix
eihk1,ωiτ0

ΩZ,f =

...

eihk1,ωiτ d





··· eihkα,ωiτ0
...
··· eihkα,ωiτ d

...

S1
| ˆF(k)|.

(4.8)

comprised of vectors ukj

as its columns. Observe that for each t ∈ R,

SWd,τ SZ f(t) = ΩZ,f · xZ,f(t).

When we go from XZ,f to SWd,τ SZ f, the toroidal geometry is minimally perturbed if the
columns ukj
of ΩZ,f are mutually orthogonal. Since α > N typically, it is not possible

99

for all pairs to be orthogonal to each other. However, we can make these as orthogonal as
possible by controlling their inner products. Suppose the embedding dimension d has been
chosen. For any two k, l ∈ Supp( ˆF), the inner product huk, uli is computed as

huk, uli = 1 + eiτhk−l,ωi + ··· + eiτhk−l,ωid.

The mutual orthogonality is maximized when the following objective function G : R → R
defined as

G(x) = X

(cid:12)(cid:12)(cid:12)1 + eixhk−l,ωi + ··· + eixhk−l,ωid(cid:12)(cid:12)(cid:12)2

k6=l

G(x). There are two obstacles
is minimized and ideally, we would like to choose τ = arg min
x
in solving this optimization problem. The first, which could be considered a blessing in
disguise is that there might not be a global minimum, that is when there is a sequence of
local minimas converging to G(x) = 0. The second is when the global minimum exists, but is
too large to be ever useful in applications. The problem then reduces to finding a reasonable
local minima, where the definition of “reasonable” depends on the size and nature of the
data. In general, one can use peak finding algorithms to detect these local minima. For the
purpose of this dissertation, we use the findpeaks(·) function in MATLAB. We add one more
condition for τ to satisfy: G(τ) < (d + 1)2. This condition is sufficient to argue that no
two vectors uk, ul will be parallel two each other, and will allow us to prove the structure
theorem. Next, we provide an example for determining a good choice of τ:

√
Example 4.3.4. Let the quasiperiodic function of interest be f(t) = eit + ei
Clearly, the set {1,
one can write it as

3+1)t.
√
3} is not incommensurate and in the language developed above,

√
3t + ei(

3, 1+

√

√
f(t) = eih(1,0),(1,

√
3)it + eih(0,1),(1,

√
3)it + eih(1,1),(1,

3)it.

Then the objective function corresponding to this f is

G(x) =

√

(cid:12)(cid:12)(cid:12)(cid:12)1 + eix
(cid:12)(cid:12)(cid:12)(cid:12)1 + eix(

+

√

33(cid:12)(cid:12)(cid:12)(cid:12)2

+(cid:12)(cid:12)(cid:12)1 + eix + ··· + eix3(cid:12)(cid:12)(cid:12)2
3−1)3(cid:12)(cid:12)(cid:12)(cid:12)2

3 + ··· + eix
√

√
3−1) + ··· + eix(

100

where we have chosen the dimension d = 3, the reasons for which will be clear in Theorem
4.3.5. The graph of the objective function G in Figure 4.2 suggests several local minimas at
x ∈ [0, 40]. To estimate these, we use the findpeaks(·) function in MATLAB on −G(x). While
in practice we will only concern ourselves in the global minima in the chosen range [0, 40],
we will analyze the top few choices for the purpose of this example. To make the process
faster, one can insert a threshold on the function −G(x). In table 4.1, we find all peaks of
−G that are higher than −0.5, i.e. all local minimas lower than 0.5, along with angles θk,l
(in degrees) between uk and ul for each of the values of x.

Figure 4.2: Example: The objective function G(x) for finding an optimal τ, for the function
√
f(t) = eit + ei

3t + ei(

√

3+1)t.

x
4.5623
23.5718
28.1141
29.9150

G(x)
0.3204
0.0056
0.2717
0.0791

θ(1,0),(1,1)
88.0983
89.6226
89.9883
89.0968

θ(0,1),(1,1)
84.4038
89.5993
85.4674
87.0751

θ(1,0),(0,1)
84.4327
89.0784
84.0524
87.3789

Table 4.1: Example of τ estimation: Candidates for τ, along with the corresponding local
minimums, and the angles θk,l (in degrees) between uk and ul

Clearly, the best choice for τ is 23.5718 out of the 4 values in the table. However, if in a
scenario, it is too big to be useful, then one would have to use τ = 4.5623. We can also see
that for all x in the table, G(x) (cid:28) (d + 1)2 = 16.

101

Having established how to choose the delay parameter τ, we now prove the main result

of this section, that is, the structure theorem:

Theorem 4.3.5. The sliding window embeddingn

SWd,τ SZ f(z) | z ∈ Zo is dense in a space

homeomorphic to TN in Td+1 if d + 1 ≥ α and if τ satisfies that G(τ) < (d + 1)2.

and let A be the L × L upper left block of ΩZ,f. Since rank(cid:16)ΩZ,f

Proof. Assume on the contrary that the matrix ΩZ,f is not full rank. Let L = min (d + 1, α)
< L, we obtain
that det(A) = 0. This implies that the rows of A are linearly dependent, so for some
c0, . . . , cL−1 ∈ C, not all zero, we have
L−1X

(cid:17)

cleihkj ,ωiτ l = 0

l=0

for all 1 ≤ j ≤ L. In other words, for each 1 ≤ j ≤ L, eihkj ,ωiτ is a root of the polynomial
P(z) = c0 + c1z + ··· + cL−1zL−1. By remark 4.2.7, all hkj, ωi are distinct. We claim that
the assumption on τ implies that all hkj, ωiτ are distinct modulo 2π, because if for some
ki 6= kj, the following is true

eihki,ωiτ = eihkj ,ωiτ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:28)

(cid:29)(cid:12)(cid:12)(cid:12)(cid:12)2

uki

, ukj

then we would obtain G(τ) ≥
= (d + 1)2 which would contradict our as-
sumption. This implies that P(z) has L distinct roots, and this further contradicts that P
is a polynomial of degree L − 1. Therefore, A is invertible and hence, ΩZ,f has full rank.
Concluding, if the embedding dimension d + 1 ≥ α, then the sliding window embedding
SWd,τ SZ f(Z) is dense in a space homeomorphic to TN embedded inside Td+1.

We would like to remark that the condition on τ in the above theorem only guarantees
the topology of a N-torus. Preserving the geometric structure as much as possible requires
choosing τ by optimizing G.

102

4.4 Persistent Homology

In this section, we present results about the Rips persistent homology of the sliding

window embedding. The strategy we use is to construct a composite Lipschitz map

Ω−1
Z,f−−−→ XZ,f

π−→ TN

SWd,τ SZ f

where the map π : XZ,f → TN is a projection onto a standard N-dimensional torus given
by




π

ˆF(kN)eihkN ,ωit
ˆF(kN+1)eihkN+1,ωit

=

ˆF(k1)eihk1,ωit

...

...

ˆF(kα)eihkα,ωit






eihk1,ωit

...

eihkn,ωit

...

eihkN ,ωit



where after reordering {k1, . . . , kN} is a linearly independent set over Q. We will first show
that π is a Lipschitz function:

Proposition 4.4.1. The projection map π

π :(cid:16)XZ,f ,||·||∞

(cid:17) →(cid:16)TN ,||·||∞

(cid:17)

is Lipschitz, with Lipschitz constant

1

| ˆF(k)min| where | ˆF(k)min| := min
k∈Supp( ˆF)

(cid:12)(cid:12)(cid:12) ˆF(k)(cid:12)(cid:12)(cid:12).

103

Proof. For any two s, t ∈ Z, we have

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)π

(cid:16)

xZ,f(t)(cid:17) − π

(cid:16)

xZ,f(s)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∞ = max

1

(cid:12)(cid:12)(cid:12)eihkn,ωit − eihkn,ωis(cid:12)(cid:12)(cid:12)
| ˆF(k)min|| ˆF(k)min|(cid:12)(cid:12)(cid:12)eihkn,ωit − eihkn,ωis(cid:12)(cid:12)(cid:12)
| ˆF(k)|(cid:12)(cid:12)(cid:12)eihkn,ωit − eihkn,ωis(cid:12)(cid:12)(cid:12)
| ˆF(k)|(cid:12)(cid:12)(cid:12)eihkn,ωit − eihkn,ωis(cid:12)(cid:12)(cid:12)

k1,··· ,kN
= max
k1,··· ,kN
| ˆF(k)min| max
k1,··· ,kN
| ˆF(k)min| max
k∈IN
Z

≤

≤

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)xZ,f(t) − xZ,f(s)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∞ .

1

1

1

=

This finishes the proof.

| ˆF(k)min|

It is important to note that the Lipschitz constant

linearly independent vectors {k1,··· , kN}.

1

| ˆF(k)min| is independent of choice of

Next, we will use theorem 3.4.3 and proposition 4.4.1 to prove a lower bound on the 1-
dimensional Rips persistence of XZ,f. Assume without a loss of generality that {k1,··· , kN}
is a linearly independent set over Q. Hence, for all 1 ≤ j ≤ α, there exist rational coefficients
cj,n ∈ Q such that

NX

n=1

kj =

cj,nkn

and consequently,

βj := hkj, ωi =

* NX

n=1

+

NX

n=1

=

cj,nkn, ω

cj,nhkn, ωi =:

NX

n=1

cj,nβn.

Define a map from TN to TαˆF

:

NP

n=1

i

where zj = ˆF(kj)e

φ : TN → TαˆF

(eiβ1t1,··· , eiβN tN ) → (z1, . . . , zα)

cj,nβntn.

104

Lemma 4.4.2. If the prime p in Zp satisfies
1
c2,n

  1

max
n

lcm

c1,n

,

!

< p,

,··· ,

1
cα,n

then there will be at least N bars of the type

3| ˆF(k)min| in bcdR
1

where ρn ≥ √
Proof. First, setting t2 = ··· = tN = 0 yields a closed curve

(cid:17).

(0, ρ1), (0, ρ2), . . . , (0, ρN)

(cid:16)XZ,f ,||·||∞
(cid:18) ˆF(k1)eiβ1t1, . . . , ˆF(kα)eicα,1β1t1
(cid:18)

(cid:19)

γ1(t1) =

(cid:19)

< δ. For all 0 ≤ m ≤ M, |sm − sm+1| < δ. This implies that

||γ1(sm+1) − γ1(sm)||∞ = max
≤ max

j

j

≤ max

j

| ˆF(kj)eiβ1cj,1sm+1 − ˆF(kj)eiβ1cj,1sm|
| ˆF(kj)β1cj,1(sm+1 − sm)|
| ˆF(kj)cj,1β1|δ

where the last term is less than 
1 by equation 4.9. We obtain a 1-homology cycle in R
1
given by

˜ν1 = [γ1(s0), γ1(s1)] + [γ1(s1), γ1(s2)] + ··· + [γ1(sM), γ1(s0)]

105

(cid:17)

(cid:16)TαˆF

. The time period of this curve is 2πQ1

∈ N, and
in TαˆF
the least common multiple is taken over reciprocals of all non zero cj,1. Let δ, 
1, 
2 > 0 be
such that

β1 where Q1 = lcm

c2,1 ,··· ,
1, 1

1
cα,1

(cid:12)(cid:12)(cid:12) δ < 
1 < 
2 <

√

3.

(4.9)

(cid:12)(cid:12)(cid:12) ˆF(kj)cj,nβn
(cid:21)

0 ≤ max
(cid:20)

j,n

of the form

2πQ1
β1
T = {0 = s0 < s1 < ··· < sM}

Let there be a finite T ⊂

0,

(cid:18)

T,

(cid:20)

0,

(cid:21)(cid:19)

2πQ1
β1

satisfying dH

and therefore, we obtain a class π∗([˜ν1]) ∈ H1

| ˆF(k)min|
Now, let {0 = u0 < u1 < ··· < uM0} := T mod 2π


1

β1 and χ1 be a closed curve in TN

R

(cid:16)TN(cid:17).

defined as

(cid:20)
0, 2π
β1

for u ∈
4.4.1,

(cid:21)
. Let ˜m be such that u ˜m corresponds to sm modulo 2π

χ1(u) = (eiβ1u, 1,··· , 1)

β1 . Then by Proposition

||χ1(u ˜m+1) − χ1(u ˜m)||∞ ≤ ||˜πγ1(sm+1) − ˜πγ1(sm)||∞ ≤

1

| ˆF(k)min| 
1.

This defines a 1-dimensional cycle µ1 in R

(TN) given by


1

| ˆF(k)min|

µ1 = [χ1(u0), χ1(u1)] + [χ1(u1), χ1(u2)] + ··· + [χ1(uM0), χ1(u0)]

satisfying π∗([˜ν1]) = Q1[µ1]. By the assumptions on the prime p, Q1 is invertible in Fp and
hence, ι∗π∗([˜ν1]) 6= 0. From the commutativity of the diagram
(cid:16)

(cid:16)

ι∗

H1

R
1

(cid:16)
R

φ
π∗

(cid:16)TN(cid:17)(cid:17)(cid:17)
(cid:16)TN(cid:17)

H1


1

| ˆF(k)min|

H1

R
2

(cid:16)
R

φ
π∗

(cid:16)TN(cid:17)(cid:17)(cid:17)
(cid:16)TN(cid:17)


2

| ˆF(k)min|

ι∗

H1

(4.10)

we obtain that ι∗([˜ν1]) 6= 0. By denseness of XZ,f in φ(TN), we can find α > 0 and points
xZ,f(t0),··· , xZ,f(tM) ∈ XZ,f such that

and

||γ1(sm) − xZ,f(tm)||∞ < α

||γ1(sm+1) − γ1(sm)||∞ + 2α < 
1.

By an application of triangle inequality, we obtain

||xZ,f(tm+1) − xZ,f(tm)||∞ < 
1

106

and this gives us another 1-homology cycle

φ

(cid:16)XZ,f

(cid:16)TN(cid:17)(cid:17) and R
1

ν1 = [xZ,f(t0), xZ,f(t1)] + [xZ,f(t1), xZ,f(t2)] + ··· + [xZ,f(tM), xZ,f(t0)]
(cid:16)

(cid:17). We claim that ν1 is just homologous to ˜ν1. If we

in both R
1
join each edge [γ1(sm), xZ,f(tm)] as well, we get ˜ν1 and ν1 as top and bottom faces of
a 2-dimensional prism of height α. Through subdivision of simplices, as in [39, Theorem
2.10], we finish the argument. It is also worth noting that a different ν satisfying the same
properties but built with different values of t0,··· , tM will be homologous to ν1 as well.

(cid:12)(cid:12)(cid:12) ˆF(kj)cj,nβn

(cid:12)(cid:12)(cid:12) ˆF(kj)cj,nβn

(cid:12)(cid:12)(cid:12) δ and all 
2 <

(cid:12)(cid:12)(cid:12) δ and ρ1 ≥ √

This implies that ι∗([ν1]) 6= 0. Therefore, it yields a bar [‘1, ρ1) such that ‘1 ≤ 
1 and
3| ˆF(k)min|, we
ρ1 ≥ 
2. Since this is true for all 
1 > max
3| ˆF(k)min|. This entire process works for all δ
get that ‘1 ≤ max
satisfying the equation and therefore, as we let δ → 0, we obtain a class alive for all 
1 > 0.
Similarly, we can obtain N 1-homology classes [ν1], . . . , [νN] that map to Q1[µ1], . . . ,
QN[µN] respectively under π∗, and correspond to bars (0, ρn) such that ρn ≥ √
3| ˆF(k)min|
for all n.

√

j,n

j,n

We will now use Lemma 4.4.2 along with cup products [39, Chapter 3] and the isomor-

i

phism of persistent homology and cohomology [24] to prove the following result.

Theorem 4.4.3. For i ≤ N, there will be at least (cid:16)N
(cid:17) bars of the type
(0, ρ1), (0, ρ2), . . . , (0, ρ(N
i ))
(cid:16)XZ,f ,||·||∞
where ρn ≥ √
Proof. By Lemma 4.4.2, ∃ N distinct 1-dimensional Rips persistent homology classes which
take birth exiting 0 and die entering ρn ≥ √
3| ˆF(k)min|. By the isomorphism of persistent
classes corresponding to (0, ρn) respectively. Pick any 
 ∈(cid:16)0,
homology and cohomology [24], there are N distinct 1-dimensional persistent cohomology

3| ˆF(k)min|(cid:17). Then we get N

3| ˆF(k)min| in bcdR

(cid:17).

√

i

1-dimensional cohomology classes

[ϑ1],··· , [ϑN]

107

in H1(cid:16)

R


(cid:16)XZ,f

(cid:17)(cid:17). Using cup products and any choice of i of these N classes, we can get

i-th dimensional classes

Varying 
 and using that the cup product is functorial, we get

H1(cid:16)
H1(cid:16)

R
1

R
2

].

[ϑl1] ^ ··· ^ [ϑli
(cid:16)XZ,f
(cid:17)(cid:17)
(cid:16)XZ,f
(cid:17)(cid:17)

R
1

(cid:16)XZ,f
(cid:17)(cid:17) × ··· × H1(cid:16)
(cid:17)(cid:17)
(cid:16)XZ,f
(cid:17)(cid:17) × ··· × H1(cid:16)
(cid:17)(cid:17)
ρn and die entering 0. Therefore, there are at least (cid:16)N

(cid:16)XZ,f
(cid:16)XZ,f

Hi(cid:16)
Hi(cid:16)

R
2

R
2

R
1

(4.11)

is commutative. This implies that the i-th dimensional cohomology classes are alive at
√
3| ˆF(k)min| ≤ min
cohomology classes in dimension i with cohomological death at 0 and persistence at least
√
3| ˆF(k)min|. The isomorphism between persistent homology and cohomology finishes the
proof.

(cid:17) persistent

n

i

We now use linear algebra to argue that Ω−1
Z,f is the conjugate transpose of ΩZ,f
ΩH

Z,f is Lipschitz. In the proposition below,

Proposition 4.4.4. Let λmin and λmax be the minimum and the maximum eigenvalues of
Z,f · ΩZ,f respectively. Then
ΩH
q(d + 1)λmax
1

Z,f · v(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)∞ ≤ 1√

||v||2 ≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Ω−1

||v||2

λmin

for each v ∈ Cα.

Z,f is Lipschitz with the constant

Therefore, the composition map π ◦ Ω−1
Theorem 4.4.5. For i ≤ N, there will be at least (cid:16)N
(cid:17) bars
(0, ρ1), (0, ρ2), . . . , (0, ρ(N
i ))
(cid:17).
SWd,τ SZ f,||·||2

where ρn ≥ √

3| ˆF(k)min|√

λmin in bcdR

(cid:16)

i

i

√

1

λmin| ˆF(k)min|.

108

Proof. By Proposition 4.4.4, we have the following commutative diagram for each i

Hi

√
(d+1)λmax
λmin

R√
R√

Hi

(cid:16)


1

SWd,τ SZ f,||·||2

(cid:16)XZ,f ,||·||∞


1

(d+1)λmax

λmin

(cid:17)
(cid:17)

(4.12)

Hi

SWd,τ SZ f,||·||2

R
1

(cid:16)
(cid:16)
R 
1√

Hi

λmin

(cid:16)XZ,f ,||·||∞

(cid:17)(cid:17)

(cid:17)

(cid:16)

where the downward maps are induced by Ω−1

ΩZ,f. By [53, Lemma 2.3] and Theorem 4.4.3, there are at least (cid:16)N

i

(cid:17) bars with length at

Z,f and the upper diagonal map is induced by

√

3| ˆF(k)min|√

least

λmin in bcdR

i

SWd,τ SZ f,||·||2

(cid:17).

4.5 Computational Example

In this section, we apply the above established theory to quasiperiodic signals. We
will first work through the entire pipeline for the special case where α = N, i.e. when
the quasiperiodic function is just the sum of exponential functions with non-commensurate
frequencies:

f(t) = eiω1t + ··· + eiωN t.

In general, we begin with a signal sampled from f, use Fast Fourier Transform to estimate
the frequencies, which we further use to estimate a good choice of τ as discussed in section
4.3, compute the sliding window point cloud and finally, compute Rips persistent homology
of SWd,τ f.

One of the main challenges in computing bcdR

i (SWd,τ f(T); F) for a quasiperiodic time
series f is that if SWd,τ f fills out the torus at a slow rate — requiring a potentially large
T ⊂ Z — then the persistent homology computation becomes prohibitively large. To combat
this, we will use an alternate strategy leveraging a persistent Künneth formula introduced in
[35]. The strategy is based on the categorical persistent Künneth formula (Corollary 3.2.6).

109

For our second example, we apply our theory to music. We start with a real music
sample with dissonance, which is a characterization of simultaneous tones that causes un-
pleasantness. We show using the developed theory that dissonance can be characterized as
quasiperiodicity in the signal.

4.5.1 An approximation strategy using a persistent Künneth formula

Let us revisit the Künneth approximation strategy from [35] and its comparison with the
more direct technique using landmark approximation which we describe first:

Landmarks: We select a set L ⊂ SWd,τ f of landmarks using maxmin sampling. That is,
‘1 ∈ SW d,τ f is chosen at random and the rest of the landmarks are selected inductively
as

minnkx − ‘1k2, . . . ,kx − ‘jk2

o

‘j+1 = argmax
x∈SW d,τ f

We endow L with the Euclidean distance in Cd, and compute the Rips barcodes
bcdR

i (L,k · k2), for i = 0, 1,··· , N, directly using Ripser [4].

We describe the Künneth strategy next:

Künneth: For 1 ≤ n ≤ N, Let pn : CN −→ C be the projections onto the n-th coordinate.
Let SWd,τ f(t) = ΩZ,f · φ(t). We select N sets of landmarks, Xn ⊂ pn ◦ φ(T) with l
points each using maxmin sampling, and endow the Cartesian product X1 × ··· × XN
with the maximum metric in CN. The barcodes of Xn, for each n are computed
using Ripser[4], and we deduce bcdR
i (X1 × ··· × XN ,k · k∞) for i = 0, 1, . . . , N using
Corollary 3.4.2.

110

4.5.1.1 An example with 2 frequencies
The general strategy will be illustrated with a computational example: Let c1, c2 ∈ C (cid:114) {0}
with |c1|2 + |c2|2 = 1, and let ω ∈ R (cid:114) Q. If

f(t) = c1eit + c2eiωt

then SWd,τ f(t) = Ω · φ(t), where

Ω =

1√
d + 1



1

1

eiτ
...
eidτ

eiωτ
...
eidωτ

 ,

√
d + 1

φ(t) =

 c1eit

c2eiωt

 .

An elementary calculation shows that if d ≥ 1 and τ =
(d+1)|ω−1|, then the columns of Ω
are orthonormal in Cd+1. Fix d = 1, ω =
2. Figure 4.3
below shows the real part of f (left), and a visualization (right) of the sliding window point
cloud SWd,τ f = SWd,τ f(T), T = {t ∈ Z : 0 ≤ t ≤ 4, 000}, using Principal Component
Analysis (PCA) [44].

√
√
3, τ as above, and c1 = c2 = 1/

2π

Figure 4.3: (Left) real(cid:16)

f(t)(cid:17) = 1√

2 cos(t) + 1√

√
2 cos(

and (Right) a PCA projection into R3 of the point cloud SWd,τ f.

3t) for the quasiperiodic time series f,

Computing the exact barcodes bcdR

i (SWd,τ f), i = 0, 1, 2, for point clouds of this size is
already prohibitively large (e.g., with Ripser [4]). Thus, one strategy is to choose a smaller

111

050100150200250300-2-1012200-21.50-1.5set of landmarks L ⊂ SWd,τ f and use the stability theorem [19]

(cid:16)bcdR

dB

i (SWd,τ f)(cid:17) ≤ 2dGH(L, SWd,τ f)

i (L), bcdR

to estimate the barcodes of SWd,τ f using those of L, up to an error (in the bottleneck
sense) of twice the Gromov-Hausdorff distance dGH between L and SWd,τ f. Specifically,
if r = dGH(L, SWd,τ f) and I = [‘I , ρI) ∈ bcdR
i (L) satisfies ρI − ‘I > 4r, then there is a
unique J ∈ bcdR

i (SWd,τ f) with |ρI − ρJ|,|‘I − ‘J| ≤ 2r, and thus

max{0 , ρI − 2r} ≤ ρJ ≤ ρI + 2r
max{0 , ‘I − 2r} ≤ ‘J ≤ ‘I + 2r.

(4.13)

We next apply the two strategies discussed above:

Landmarks: We select a set L ⊂ SWd,τ f of 400 landmarks (%10 of the data) using maxmin
sampling. We endow L with the Euclidean distance in C2, and compute the barcodes
bcdR

i (L,k · k2), i = 0, 1, 2, directly using Ripser [4].

Künneth: We select two sets of landmarks, XL ⊂ p1 ◦ φ(T) and YL ⊂ p2 ◦ φ(T) with 70
points each using maxmin sampling, and endow the cartesian product XL × YL with
the maximum metric in C2. The barcodes of XL and YL are computed using Ripser,
and we deduce those of bcdR

i (XL × YL,k · k∞) for i = 0, 1, 2 using Corollary 3.2.6.

4.5.1.2 Computational Results

Table 4.2 shows the computational times in each approximation strategy — i.e., via the
landmark set L and the persistent Künneth theorem applied to XL × YL — as well as the
number of data points in each case.

As these results show, the Künneth approximation strategy is almost two orders of mag-
nitude faster than just taking landmarks, and as we will see below, it is also more accurate.

112

Landmarks Künneth

Time (sec)
# of points

15.22
400

0.20
4,900

Table 4.2: Computational times, and number of points, for the landmark and Künneth
approximations to bcdR

i (SWd,τ f), i ≤ 2.

Indeed, the inequalities in (4.13) can be used to generate a confidence region for the ex-
istence of an interval J ∈ bcdR
i (L). A
similar region can be estimated from bcdR
i (XL × YL) as follows. Since the columns of Ω are
orthonormal, then for every z ∈ C2 we have kzk∞ ≤ kzk2 = kΩzk2 ≤ √

i (SWd,τ f) given a large enough interval I ∈ bcdR

2kzk∞ and thus

R
(SWd,τ f,k · k2) ⊂ R
(φ(T),k · k∞) ⊂ R√

2
(SWd,τ f,k · k2).

Moreover, (see 3.1.21) if I ∈ bcdR
i (XL × YL,k · k∞) satisfies
− √

XL × YL, φ(T)(cid:17)

2‘I > 4dGH

(cid:16)

ρI√
2

where λ = dGH(XL × YL, φ(T)) is computed for subspaces of (C2,k · k∞), then there exists
a unique J ∈ bcdR

i (SWd,τ f,k · k2) so that
)
ρI − 2λ√
)
2
‘I − 2λ√
2

(
(

max

0 ,

max

0 ,

≤ ρJ ≤ √
≤ ‘J ≤ √

2 · (ρI + 2λ)

2 · (‘I + 2λ).

(4.14)

Figure 4.4 shows the resulting confidence regions for both approximation strategies. Each
i (L) (4.13) and
i (XL × YL) (4.14) are shown in blue and red, respectively. We compare these regions

interval [‘, ρ) is replaced by a point (‘, ρ) ∈ R2, and the regions for bcdR
bcdR
by computing their area, and report the results in Table 4.3.

i Landmarks Künneth
1
1
2

0.4680
0.4709
0.4704
i (L) (blue) and bcdR
Table 4.3: Areas for confidence regions from bcdR
i = 1, 2 (see Figure 4.4). A smaller area suggests a better approximation.

0.7677
0.7480
0.9072

i (XL × YL) (red),

113

Figure 4.4: Confidence regions for bcdR
i (XL × YL), i = 1, 2. For a given
color, each box (confidence region) contains a point (‘J , ρJ) corresponding to a unique
J ∈ bcdR

i (L) and bcdR

i (SWd,τ f).

These results suggest that the Künneth strategy provides a tighter approximation than
using landmarks. Moreover, both strategies can be used in tandem — improving the quality
of approximation — via intersection of confidence regions.

4.5.2 An application to music theory

In music theory, consonance and dissonance are classification of multiple simultaneous tones.
While the former is associated with pleasantness, the latter creates tension as experienced
by the listener. Perfect dissonance occurs when the audio frequencies are irrational with
respect to each other. One of such instances is that of a tritone, which a musical interval
that is halfway between two octaves. Mathematically, for a base frequency ω1, the tritone is
√
2ω1. We will use the theory of sliding window embedding to quantify quasiperiodicity from
a dissonant sample. For the purpose of this application, we use a 5-second audio recording
of a brass horn provided by Adam Huston. It was read by audioread function in MATLAB
and the signal plot is shown in Figure 4.5.

114

Figure 4.5: The audioread plot of a dissonant music sample created on a brass horn.

Like before, in order to perform sliding window analysis, we need to choose appropriate
parameters d and τ. The first step is to use Fast Fourier Transform in MATLAB to obtain
the amplitude-frequency spectrum. We use findpeaks with thresholds MinPeakHeight = 0.06

Figure 4.6: The amplitude-frequency spectrum of the music sample plot using Fast Fourier
Transform.

and MinPeakDistance = 100 to detect frequencies which we will use for estimation of the
embedding parameters. See Table 4.4.

Angular Frequencies

Frequencies (Hz)

Proportion

1384.93
220.41

1

1957.83
311.59
1.4137 ≈ √
2

2769.86
440.83

2

3911.93
622.60
√
2.8246 ≈ 2

2

Table 4.4: List of frequencies in the amplitude-frequency spectrum: The first row is the list
of frequencies that were detected. The second row is their conversion to Hertz. The third
row is their ratio with respect to the first one.

115

Remark 4.5.1. While our entire theory has been developed for complex functions, it can be
adjusted to work with real functions via the following identities:

cos(at) = 1

2 eiat + 1

2 ei(−a)t

and

sin(at) = 1
2i

eiat − 1
2i

e−i(−a)t

for any a ∈ R, t ∈ R.

For the purpose of our analysis, this would mean that instead of 4 frequencies, we count
8 of them (with their negatives). By the theory, the complex embedding dimension d should
be at least 8, that would imply that the real embedding dimension should be at least 8.
Therefore, we choose the embedding dimension d = 9, and the delay τ = 0.02858 in accor-
dance with the optimization discussion in section 4.3. We use cubic splines to compute the
sliding window vectors and present the PCA representation of the point cloud, along with
the persistence diagrams in Figure 4.7. We conclude the dissonance is indeed a quasiperiodic
feature.

Figure 4.7: (Left) PCA representation of the sliding window point cloud generated from the
dissonant sample with parameters d = 9 and τ = 0.02858. (Right) Persistence Diagrams in
homological dimension 1 and 2. The dotted lines separate the top 2 persistence points (sim.
maximum persistence point) from the rest in dimension 1 (sim. dimension 2).

116

CHAPTER 5

STABILITY OF MULTISCALE CIRCULAR COORDINATES

In this chapter, we provide a stability characterization of circular coordinates. We set up
the problem of stability with respect to the choice of landmarks in section 5.1. In section
5.2, we present stability results for the geometric noise model. In section 5.3, we approach
the general case and prove stability for the Wasserstein noise model. The last section 5.4
is dedicated to examples. Parts of this chapter are to appear in [34]. We begin with a
description of the sparse circular coordinates algorithm [55, Section 1.2] first.

5.0.1 The sparse circular coordinates algorithm

1. Let (M, d) be a finite metric space (i.e., the dataset) and select a set of landmarks

L = {‘1, . . . , ‘N} ⊂ M, e.g., randomly or via maxmin sampling.

2. Choose a prime q > 2 and compute the 1-dimensional Rips persistent cohomology
P H1(R(L); Zq) on (L, d|L) with coefficients in Zq. Let bcd(L) denote the resulting
persistence diagram.

3. Let rL = max

x∈M min
‘∈L

d(x, ‘) be the minimal cover radius. Observe that rL = dH(L, M).

If there exists (a, b) ∈ bcd(L) such that max{a, rL} < b2, then let

α = t · max{a, rL} + 1

2(1 − t)b

for some t ∈ (0, 1) and η00 ∈ Z1(R2α(L); Zq) be a cocycle representative for the persis-
tent cohomology class corresponding to (a, b).

4. Lift η00 to an integer cocycle η0 ∈ Z1(R2α(L); Z) such that η00−η0 mod q is a cobound-

ary in C1(R2α(L); Zq). In practice, we use the cochain

η0(σ) =

η00(σ)

η00(σ) − q

117

if η00(σ) ≤ q−1
2
if η00(σ) > q−1
2 .

Lastly, define η = ι1η0 where ι1 : C1(R2α(L); Z) → C1(R2α(L); R) is the homomor-
phism induced by the inclusion ι : Z ,→ R.

5. Choose positive weights for the vertices and edges of R2α(L) and let

2α : C1(R2α(L); R) → C0(R2α(L); R)
d+

be the weighted Moore-Penrose pseudoinverse of the coboundary map

d2α : C0(R2α(L); R) → C1(R2α(L); R).

and define

τ = −d+

2α(η)

and

θ = η + d2α(τ).

6. Denote τj := τ([‘j]) and θjk := θ([‘j, ‘k]) when [‘j, ‘k] ∈ R2α(L) (it is zero otherwise).
Let Bα(‘j) denote a ball of radius α centered around ‘j ∈ L and {ϕj}j denote the par-
tition of unity subordinate to the collection {Bα(‘j)}j. The sparse circular coordinates
are defined as follows:

N[

hθ,τ :

Bα(‘j) → S1 ⊂ C

j=1
Bα(‘j) 3 b 7→ exp

2πi

τj +

NX

r=1

 .

ϕr(b)θjr

The circular coordinates algorithm can be used to reconstruct dynamical systems as in the
following example:

√

2 ei

2 eit + 1√

Example 5.0.1. The quasiperiodic function f(t) = 1√
3t in section 4.5.1.1 can
be realized as an observation of the irrational winding on a 2-torus starting at (0, 0), with
√
3. The theory developed in the last chapter tells that the sliding window embedding
slope
of the function will be a 2-torus and persistence verifies that. Using the two 1-persistent
cocycles, we can generate two sets of circular coordinates using the algorithm above, and
together we reconstruct an approximation to the original winding. See figure 5.1 for the
entire pipeline.

118

√

Figure 5.1: An example of reconstructed toroidal dynamics: (Left) The irrational winding
3. (Second) The sliding window embedding of the
on a 2-torus starting at (0, 0) with slope
observation f(t) = 1√
3t of the winding. (Third) The Rips persistence diagram
in dimension 0, 1. The two 1-persistent cocycles can be used to create two sets of circular
coordinates. (Right) The pair of coordinates can be used to reconstruct the toroidal winding.
The average slope is estimated to be 1.9698.

2 eit + 1√

√

2 ei

Note that the algorithm depends on a choice of landmarks L ⊂ M and as one would
expect, different choices of landmarks may yield different circular coordinates. One would
also expect that two landmark sets that are “close” to each other in some sense yield circular
coordinates that are also “close” in some sense. In the next section, we set up the problem
of stability.

5.1 The stability set-up

We start with two finite landmark sets L, ˜L of the metric space (M, d) with probability

weights ω and ˜ω satisfying that

dW,∞((L, ω), (˜L, ˜ω)) < δ

where dW,∞ is the Wasserstein-Kantorovich-Rubinstein metric as in Definition 2.5.5. The
goal is to obtain results on the proximity of the associated circular valued functions in the
L∞-metric. We first solve the problem for a special case, that is when L and ˜L are in a
weight preserving bijective correspondence: We call this the Geometric Noise Model. We
approach the general case by duplicating some landmarks in a well-defined manner to create
a bijection and therefore, reducing it to the Geometric Noise Model. Next, we give explicit
choices for partitions of unity and edge weights that we use in our stability analysis.

119

200-21.50-1.55.1.1 Partition of Unity

Let φ : R → [0,∞) be increasing and Lipschitz, i.e. for all a, b ∈ R, φ(a) ≤ φ(b) if a ≤ b,
and |φ(a) − φ(b)| ≤ Lφ |a − b| where Lφ is the Lipschitz constant. Moreover, let φ(a) = 0
for all a ≤ 0.

For an α > 0, and the function φ defined above, the collection of functions n
Proposition 5.1.1. Let L = {‘1, . . . , ‘N} ⊂ (M, d) be a finite set with weights {ω1, . . . , ωN}.

ϕj : M → Ro

defined as

ϕj(b) = ωjφ(α − d(b, ‘j))
ωkφ(α − d(b, ‘k))

NP
Bα(‘j)oN

k=1

is a partition of unity subordinated to n
Proof. By construction, we have NP
ϕj(b) = 1. Thus, we only need to verify that Supp(ϕj) ⊂
Bα(‘j). Clearly, if b 6∈ Bα(‘j), then d(b, ‘j) ≥ α, thus α − d(b, ‘j) ≤ 0 and by construction
of φ, then φ(α − d(b, ‘j)) = 0.

j=1.

If there is another finite set ˜L = n˜‘1, . . . , ˜‘N
satisfying d(˜‘i, ‘i) < δ, then for α− δ we can define another partition of unityn

o ⊂ (M, d) with weights {˜ω1, . . . , ˜ωN}
ψj : M → Ro

j=1

as

We introduce notation:

ψj(b) =

˜ωjφ(α − δ − d(b, ˜‘j))
NP
˜ωkφ(α − δ − d(b, ˜‘k))

k=1

NX
NX

k=1

k=1

ϕ(b) =

ψ(b) =

ωkφ(α − d(b, ‘k))

˜ωkφ(α − δ − d(b, ˜‘k)).

If the weights satisfy ωk = ˜ωk for all k, then the following is true:
Proposition 5.1.2. If ωk = ˜ωk for all k, and d(˜‘k, ‘k) < δ for all k, then ψ(b) ≤ ϕ(b) for
all b ∈ M.

120

Proof. By triangle inequality, since (cid:12)(cid:12)(cid:12)d(b, ‘k) − d(b, ˜‘k)(cid:12)(cid:12)(cid:12) ≤ d(‘k, ˜‘k) < δ, then

d(b, ‘k) < δ + d(b, ˜‘k).

By properties of φ, we obtain φ(α − δ − d(b, ˜‘k)) ≤ φ(α − d(b, ‘k)) for each k. Therefore,
ψ(b) ≤ ϕ(b).

Proposition 5.1.3. For b ∈ L(α−δ), we have

NX

(cid:12)(cid:12)(cid:12)ϕj(b) − ψj(b)(cid:12)(cid:12)(cid:12) ≤ 4δLφ

ϕ(b) .

j=1
Proof. We will proceed as follows:

(cid:12)(cid:12)(cid:12)ϕj(b) − ψj(b)(cid:12)(cid:12)(cid:12) =

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

− ˜ωjφ(α − δ − d(b, ˜‘j))
NP
˜ωkφ(α − δ − d(b, ˜‘k))
(cid:12)(cid:12)(cid:12)δ + d(b, ˜‘j) − d(b, ‘j)(cid:12)(cid:12)(cid:12) +
˜ωjφ(α − δ − d(b, ˜‘j))

ωjφ(α − d(b, ‘j))
NP
ωkφ(α − d(b, ‘k))
k=1
≤ ωjLφ
ϕ(b)
≤ ωjLφ
ϕ(b) 2δ +

˜ωjφ(α − δ − d(b, ˜‘j))

ψ(b)

k=1

ψ(b)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) .

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1 − ψ(b)

ϕ(b)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

where in the second step, we added and subtracted ωj φ(α−δ−d(b,˜‘j))
ωkφ(α−d(b,‘k))

, and used triangle

inequality, while keeping in mind that ω and ˜ω are the same. Summing over all j yields

(cid:12)(cid:12)(cid:12)ϕj(b) − ψj(b)(cid:12)(cid:12)(cid:12) ≤ 2δLφ

ϕ(b)

NX

j=1

NX

j=1

ωj +

ϕ(b) +

2δLφ
ϕ(b) =

4δLφ
ϕ(b) .

ϕ(b)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1 − ψ(b)
NP
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ 2δLφ
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)1 − ψ(b)

ϕ(b)

k=1

5.1.2 Weights on edges
By a weight family on the edges of L, we mean a sequence of functions να : Rα(L)(1) → [0,∞)
(parametrized by α) satisfying the following properties:

• να is increasing in α, and

121

(cid:16)[‘i, ‘j](cid:17), which we call the weight of the edge [‘i, ‘j], is 0 if and only if 0 < α ≤

• να

d(‘i, ‘j).

For the purpose of this dissertation, we define these να in the same spirit as the partitions
of unity, i.e. controlled by an increasing and Lipschitz function ρ : R → [0,∞) satisfying
ρ(a) = 0 for all a ≤ 0, and by vertex weights ω in the following manner:

να([‘i, ‘j]) = ωiωjρ(α − d(‘i, ‘j)).

(5.1)

The vertex weights ω and the weight families να define inner products h·,·iα on C0(Rα(L); R)
and C1(Rα(L); R) respectively, by letting the indicator functions 1σ on k-simplices σ ∈
Rα(L) (for k = 0, 1) be orthogonal, and setting

h1[‘i], 1[‘i]i = ωi

and h1σ, 1σiα = να(σ).

Similarly, we define vertex weights ˜ω and a weight family ˜να on the edges of ˜L

˜να([˜‘i, ˜‘j]) = ˜ωi˜ωjρ(α − d(˜‘i, ˜‘j)).

and the induced inner products on C0(Rα(˜L); R) and C1(Rα(˜L); R) follow.

5.2 Geometric noise model

Let L = {‘1, . . . , ‘N} and ˜L = {˜‘1, . . . , ˜LN} be subsets of M such that there is a bijection

µ : ˜L → L satisfying the following conditions:

1. d(˜‘i, µ(˜‘i)) < δ

2. µ(˜‘i) = ‘i

3. ˜ωi := ˜ω(˜‘i) = ω(‘i) =: ωi

for all i ∈ {1, . . . , N}. By an application of triangle inequality,

d(˜‘i, ˜‘j) < 2α − 2δ ⇒ d(cid:16)

(cid:17)

‘i, ‘j

< 2α,

122

and therefore, µ extends to a simplicial map

µ : R2α−2δ(˜L) → R2α(L),

where α is chosen so that R2α−2δ(˜L), and hence R2α(L), is connected. In case of discon-
nectedness, one can consider each connected component separately. Consider the induced
commutative diagrams

C0(R2α−2δ(˜L); R)

˜d2α−2δ

C1(R2α−2δ(˜L); R)

µ0

µ1

C0(R2α(L); R)

d2α

C1 (R2α(L); R)

where ˜d2α−2δ, d2α are the coboundary maps, and µ0, µ1 are the maps induced by the sim-
plicial map µ. In fact, µ0 is an isometry since it preserves norms.

Let να, ˜να be the weight-families on the edges of L, ˜L respectively.

Proposition 5.2.1. If the edge weights satisfy the condition in equation 5.1, then

(cid:12)(cid:12)(cid:12)ν2α([‘i, ‘j]) − ˜ν2α−2δ([˜‘i, ˜‘j])(cid:12)(cid:12)(cid:12) ≤ 4ωiωjLρδ.

Proof. By our choice in equation 5.1 and the assumption on vertex weights, we obtain

(cid:12)(cid:12)(cid:12)ν2α([‘i, ‘j]) − ˜ν2α−2δ([˜‘i, ˜‘j])(cid:12)(cid:12)(cid:12) = ωiωj
(cid:12)(cid:12)(cid:12)ρ(2α − d(‘i, ‘j)) − ρ(2α − 2δ − d(˜‘i, ˜‘j))(cid:12)(cid:12)(cid:12) ≤ Lρ

(cid:12)(cid:12)(cid:12)ρ(2α − d(‘i, ‘j)) − ρ(2α − 2δ − d(˜‘i, ˜‘j))(cid:12)(cid:12)(cid:12) .
(cid:12)(cid:12)(cid:12)2δ + d(˜‘i, ˜‘j) − d(‘i, ‘j)(cid:12)(cid:12)(cid:12)

By Lipschitzness of ρ, we obtain

which is less than Lρ4δ by triangle inequality. This finishes the proof.

Now, let ( ˜d2α−2δ)+ and (d2α)+ be the weighted Moore-Penrose pseudoinverses

( ˜d2α−2δ)+ : C1(R2α−2δ(˜L); R) → C0(R2α−2δ(˜L); R)

(d2α)+ : C1 (R2α(L); R) → C0 (R2α(L); R)

123

of the coboundary maps ˜d2α−2δ and d2α respectively. For the the image

0 6= η ∈ Img{C1 (R2α(L); Z) → C1 (R2α(L); R)},

let

C0(R2α(L); R) 3 τ = −d+

2α(η)
C1(R2α(L); R) 3 θ = η + d2α(τ)

˜τ = −( ˜d2α−2δ)+µ1(η) ∈ C0(R2α−2δ(˜L); R)
˜θ = µ1(η) + ˜d2α−2δ(˜τ) ∈ C1(R2α−2δ(˜L); R).

We want to bound the difference between the circle valued functions

˜h : ˜L(α−δ) → S1
Bα−δ(˜‘j) 3 b → exp

(

2πi

 
˜τj +X

r

!)

ψr(b)˜θjr

and

h : L(α) → S1
Bα(‘j) 3 b → exp

(

2πi

 
τj +X

!)

ϕr(b)θjr

r

on their common domain ˜L(α−δ) ⊂ L(α), where the partitions of unity {ϕr}r and {ψr}r are
as defined above.

Lemma 5.2.2. The functions ˜h and h defined as above differ on their common domain by

|h(b) − ˜h(b)| ≤ 2π

where ||ϕ(b) − ψ(b)||1 =P

r

  φ(α − δ)
|ϕr(b) − ψr(b)|.

ψ(b)

||µ0(τ) − ˜τ||ω + ||ϕ(b) − ψ(b)||1||θ||∞

,

!

Proof. From the definitions of θ and ˜θ, we have that

(cid:17) = µ1(η)jr = ηjr = θjr −(cid:16)
˜θjr −(cid:16)˜τr − ˜τj
(cid:17), it is indeed the case and therefore,
(cid:16)
if and only if the index pair (j, r) represents an edge in both R2α−2δ(˜L) and R2α(L). If
b ∈ Bα−δ(˜‘j) ⊂ Bα
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)exp
(
(
|h(b) − ˜h(b)| =
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16)

 
τj +X
ϕr(b)θjr
(cid:16)
(cid:17) +X
ϕr(b)θjr − ψr(b)˜θjr

 
˜τj +X

!)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

ψr(b)˜θjr

2πi

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

τr − τj

− exp

≤ 2π

τj − ˜τj

!)

2πi

(cid:17)

r

‘j

r

r

124

We rewrite τj as P
ψr(b)τj. Observe that if ψr(b) 6= 0, then Bα−δ(˜‘r) ∩ Bα−δ(˜‘j). This is
equivalent to saying that the index (j, r) represents an edge in both R2α−2δ(˜L) and R2α(L).
Therefore,

r

ϕr(b)θjr − ψr(b)˜θjr

(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

|h(b) − ˜h(b)| ≤ 2π

= 2π

≤ 2π

≤ 2π

≤ 2π

(cid:17) +X

(cid:16)

r

r

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X
ψr(b)(cid:16)
τr − ˜τr + ˜θjr − θjr
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X
ψr(b) (τr − ˜τr) +X
 X
ψr(b) · |τr − ˜τr| +X
 X
ψ(b)φ(α − δ) · |τr − ˜τr| +X
 φ(α − δ)

ωr

r

r

r

r

r

r

(ϕr(b) − ψr(b)) θjr
!
|θjr| · |ϕr(b) − ψr(b)|

!
|θjr| · |ϕr(b) − ψr(b)|

!

||µ0(τ) − ˜τ||ω + ||ϕ(b) − ψ(b)||1||θ||∞

.

ψ(b)

where the last step comes from Hölder’s inequality.

The ‘1 bound on the difference of partitions of unity was proved in lemma 5.1.3. The
rest of this section is dedicated to finding an upper bound for ||µ0(τ) − ˜τ||ω. By definition,
we have

(˜τ − µ0(τ)) = ( ˜d2α−2δ)+µ1(η) − µ0d+

2α(η).

Using that η + d2α(τ) = θ, the commutativity relations, and the properties of the Moore-
Penrose pseudoinverse, we get

( ˜d2α−2δ)+µ1(η) = ( ˜d2α−2δ)+ (−µ1d2α(τ) + µ1 (θ))

= −( ˜d2α−2δ)+ ˜d2α−2δµ0(τ) + ( ˜d2α−2δ)+µ1 (θ)
= −PrKer( ˜d2α−2δ)⊥ (µ0(τ)) + ( ˜d2α−2δ)+µ1 (θ)

where we have used that ( ˜d2α−2δ)+ ˜d2α−2δ projects surjectively onto the orthogonal comple-
ment of Ker( ˜d2α−2δ). Using the connectedness of R2α−2δ(˜L), we get

(˜τ − µ0(τ)) = −PrKer( ˜d2α−2δ)⊥ (µ0(τ)) + ( ˜d2α−2δ)+µ1 (θ) + µ0(τ)

= PrKer( ˜d2α−2δ) (µ0(τ)) + ( ˜d2α−2δ)+µ1 (θ)
= ( ˜d2α−2δ)+µ1 (θ)

125

where the last step is true because µ0(τ) ⊥ Ker( ˜d2α−2δ).
Hence, it is enough to bound the 0-cochain ( ˜d2α−2δ)+µ1 (θ) ∈ C0(R2α−2δ(˜L); R). First,
since the Moore-Penrose pseudoinverse satisfies A+ = A+AA+ for any matrix A, we obtain

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

ω

≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

2α−2δ

(5.2)

where the first norm is the weighted operator norm. We will bound both the norms sepa-
rately. Let

( ˜d2α−2δ)∗ : C1(cid:16)

R2α−2δ(˜L); R(cid:17) → C0(cid:16)

R2α−2δ(˜L); R(cid:17)

be the adjoint of ˜d2α−2δ with respect to the inner products defined by the weights ˜ν2α−2δ
and the probability measure ˜ω = ω. Then it satisfies the following proposition:

Proposition 5.2.3. Let K be a simplicial complex with edge weights

νK : K(1) → [0,∞)

If d : C0 (K, R) → C1 (K, R) is the coboundary operator, let d∗ : C1 (K, R) → C0 (K, R) be
its adjoint corresponding to the inner product on 1−chains induced by νK, then d∗d is the
graph Hodge Laplacian on K(1).

An application of the spectral theorem to the graph Hodge Laplacian operator ∆H =
( ˜d2α−2δ)∗ ˜d2α−2δ yields an orthonormal eigenbasis ˜x1, . . . , ˜xN for C0(R2α−2δ(˜L); R) with
eigenvalues 0 = (λ1)2 < (λ2)2 ≤ ··· (λN)2. The eigenvalue 0 has multiplicity 1 because
R2α−2δ(˜L) is connected.
Let B be another orthonormal basis of C0(R2α−2δ(˜L); R) defined as

( 1√

B =

[˜‘j]

ωj

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) j = 1, . . . , N

)

.

126

Then with the respect to the basis B, the matrix LH of ∆H can be computed as

LH(j, k) =

=

=

ωj

!

+

[˜‘j]

[˜‘k]

1√
ωk

,

*
∆H
1√
ωjωk

  1√
D ˜d2α−2δ([˜‘j]), ˜d2α−2δ([˜‘k])E
P
1≤i≤N
√

˜ν2α−2δ([˜‘i, ˜‘j])

1
ωj
− ˜ν2α−2δ([˜‘j ,˜‘k])



if j = k

if j 6= k.

ω

ωj ωk

2α−2δ

Note that since change of basis doesn’t change the eigenvalues of the matrix, the smallest
non-zero eigenvalue of LH is also (λ2)2. This will be used while proving the Wasserstein
stability later in this Chapter. We are now ready to bound the operator norm.

Lemma 5.2.4. The weighted pseudoinverse of the coboundary map ˜d2α−2δ satisfies

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 =

  1

!2

λ2

Proof. Since the eigenvalues of ( ˜d2α−2δ)∗ ˜d2α−2δ are 0 = (λ1)2 < (λ2)2 ≤ ··· (λN)2, we get
that λ2 is the maximum eigenvalue of the pseudoinverse ( ˜d2α−2δ)+. Therefore,

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =

  1

!

.

λ2

Lemma 5.2.5. The weighted norm of ( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ) satisfies

Proof. First, let v = µ−1

0 ( ˜d2α−2δ)+µ1(θ) ∈ C0(R2α(L); R). Then

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

2α−2δ

2α−2δ

.

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1(θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω
≤ 4δLρ ||θ||∞
=D
µ1 (θ) , ( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)E
= hµ1 (θ) , µ1d2α(v)i2α−2δ .

2α−2δ

127

The last inner product can be computed as

˜ν2α−2δ(e)θ(e)d2α(v)(e)

e

X
=X
= X

e

1≤j<k≤N

(˜ν2α−2δ(e) − ν2α(e)) θ(e)d2α(v)(e)

(cid:16)˜ν2α−2δ

(cid:16)[˜‘j, ˜‘k](cid:17) − ν2α

(cid:16)[‘j, ‘k](cid:17)(cid:17)

θjk(vk − vj)

Taking absolute values, we obtain

(cid:12)(cid:12)(cid:12)(cid:16)|vk| +(cid:12)(cid:12)(cid:12)vj

(cid:12)(cid:12)(cid:12)(cid:17)

(cid:16)[‘j, ‘k](cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)θjk
(cid:12)(cid:12)(cid:12)(cid:17)

2α−2δ

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:12)(cid:12)(cid:12)˜ν2α−2δ
(cid:16)[˜‘j, ˜‘k](cid:17) − ν2α
≤ X
X
1≤j<k≤N
≤ 4δLρ ||θ||∞
X
X
 X

(cid:16)|vk| +(cid:12)(cid:12)(cid:12)vj
(cid:12)(cid:12)(cid:12)(cid:17)
(cid:16)(cid:12)(cid:12)(cid:12)vj
1

(cid:12)(cid:12)(cid:12)
(cid:12)(cid:12)(cid:12)vj
(cid:12)(cid:12)(cid:12)vj

≤ 4δLρ ||θ||∞

≤ 4δLρ ||θ||∞

= 4δLρ ||θ||∞

1≤j<k≤N

1≤j,k≤N

ωj

1≤j≤N

(cid:12)(cid:12)(cid:12)2

2

ωjωk

ωjωk

ωj

1≤j≤N

=4δLρ ||θ||∞ ||v||ω .

Since µ0 is an isometry, we obtain

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

2α−2δ

≤ 4δLρ ||θ||∞

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1(θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω

.

(5.3)

Now, we can put together the two lemmas to bound the difference between ˜τ and µ0(τ):

Proposition 5.2.6. The difference between cochains ˜τ and µ0(τ) is bounded by:

||(˜τ − µ0(τ))||ω ≤ kθk∞ 4Lρ
(λ2)2

δ.

128

Proof. Recall equation 5.2:

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

ω

≤(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

2α−2δ

.

By Lemmas 5.2.4 and 5.2.5, we obtain

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1 (θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2

ω

≤

  1

!2

λ2

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( ˜d2α−2δ)+µ1(θ)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω

.

4δLρ ||θ||∞

This finishes the proof since the above implies

||(˜τ − µ0(τ))||ω ≤ kθk∞ 4Lρ
(λ2)2

δ.

We now use Lemma 5.2.2 and Proposition 5.2.6 to achieve the following result:

Theorem 5.2.7. If (L, ω) and (˜L, ˜ω) are landmark sets in a weight preserving bijective
correspondence, and h and ˜h are the associated circle-valued functions, then

||h − ˜h||∞ ≤ 8πkθk∞

δ

min
j

ωj

 φ(α − δ)
(cid:16)

α − δ − r˜L

φ

(cid:17) Lρ
(λ2)2 +

Lφ

φ (α − rL)

 .

Proof. Recall that Lemma 5.1.3 gives us the bounds on partitions and therefore,

By Lemma 5.2.2 and Proposition 5.2.6, we get that

||ϕ(b) − ψ(b)||1 ≤ 4δLφ
¯ϕ(b) .
 φ(α − δ)

|h(b) − ˜h(b)| ≤ 8πkθk∞δ

ψ(b)

Lρ

(λ2)2 + Lφ
¯ϕ(b)

!

.

(5.4)

The average terms ϕ(b) and ψ(b) depend on the choice of b. Since L(α) covers M, then there
exists j such that d(b, ‘j) ≤ rL (the cover radius of L). Therefore,

φ(α − rL) ≤ φ(α − d(b, ‘j)) ≤ NX

φ(α − d(b, ‘k)).

k=1

129

Similarly, since ˜L(α−δ) also covers M,

Consequently,

NX
NX

k=1

k=1

ϕ(b) =

ψ(b) =

This implies that

ωkφ(α − d(b, ‘k)) ≥ min

φ(α − d(b, ‘k))

ωjφ(α − rL)

ωkφ(α − δ − d(b, ˜‘k)) ≥ min

φ(α − δ − d(b, ˜‘k))

ωjφ(α − δ − r˜L).

k=1

φ(α − δ − r˜L) ≤ NX
 NX
 NX
 φ(α − δ)
(cid:16)

k=1

k=1

ωj

ωj

δ

j

j

min
j

ωj

φ

α − δ − r˜L

||h − ˜h||∞ ≤ 8πkθk∞

φ(α − δ − d(b, ˜‘k)).

 ≥ min

j

 ≥ min
 .

Lφ

j

(cid:17) Lρ
(λ2)2 +

φ (α − rL)

Remark 5.2.8. Before we proceed to the general case, it is worth noting that since h and
˜h are functions taking values in a circle of unit radius, the ‘∞ distance on the left can not
exceed 2. Since the expression on the right is a real number, the bound in Theorem 5.2.7
only tells us something about stability, if

8kθk∞

min
j

ωj

 φ(α − δ)
(cid:16)

α − δ − r˜L

φ

δ ≤

(cid:17) Lρ
(λ2)2 +

Lφ

φ (α − rL)

−1


.

5.3 The general Wasserstein noise model

Let L, ˜L be finite sets with different cardinalities, say N and ˜N respectively. Also, assume
that ω and ˜ω are the probability weights on the two sets respectively. Assume that these
sets satisfy

dW,∞((L, ω), (˜L, ˜ω)) < δ.

In Section 5.3.1, we will construct sets (Lm, ωm) and (˜Lm, ˜ωm) such that they are in a weight
preserving bijection. As a result, the coordinates generated by (Lm, ωm) and (˜Lm, ˜ωm) will
be compared using bounds from the geometric stability case.

130

5.3.1 Landmark Augmentation

Let (M, d) be a metric space. Let L, ˜L ⊂ M be finite sets with N = |L| and ˜N =(cid:12)(cid:12)(cid:12)˜L

(cid:12)(cid:12)(cid:12) together

with measures ω and ˜ω on M supported in L and ˜L respectively. The goal of this section is
to build extended landmark sets Lm and ˜Lm, and measures ωm and ˜ωm supported on Lm
and ˜Lm respectively, such that Lm is in a weight preserving bijection with ˜Lm.

The first step in our construction is to provide an algorithm to extend one landmark set
in a controlled manner. The following lemma showcases the basic tools for such construction.

Lemma 5.3.1. For any matrix A ∈ Rn×m, there exists a matrix B such that

1. B has size n0 × m where n0 is the number of non-zero entries in A,

2. B has at most one non-zero entry per row, and

3. for each 1 ≤ j ≤ m, the non-zero entries of A·,j and B·,j are the same.

Proof. See [34].

Remark 5.3.2. Conditions (1) and (2) imply that there is exactly one non-zero entry per row
in B. Therefore, the number of non-zero entries in B is the same as in A.

Remark 5.3.3. The matrix B can be indexed using a block-wise construction. From here on,
we will denote by b(i,ik),j the entry of B in the k-th row of the i-th block corresponding to
the j-th column.

131



...
...

k
...
...

i-th block

··· j ···

...
...

...
. . .
··· b(i,ik),j
. . .
...

...
...



...
. . .
···
. . .
...

Let dδ be the metric on N such that for any (s, t) ∈ N2, dδ(s, t) = δs,t, where δs,t is 1 if

s = t and 0 otherwise.

Theorem 5.3.4. Let β > 0, let dβ = d + βdδ be a metric on M × N and let ω and ˜ω be
measures on (M, d) supported on L, ˜L respectively. If dW,∞((L, ω), (˜L, ˜ω)) < δ , then there
exist functions m : L → N and ˜m : ˜L → N and measures ωm and ˜ωm on (M, dβ) with
supports

Lm = {(‘, k) ∈ L × N : 1 ≤ k ≤ m(‘)}

˜Lm =n(˜‘, s) ∈ ˜L × N : 1 ≤ s ≤ ˜m(˜‘)o

such that

m(‘)P

1. ω(‘) =

ωm(‘, k) for all ‘ ∈ L and ˜ω(˜‘) =

ωm(˜‘, s) for all ˜‘ ∈ ˜L.
2. There is a bijection ϑ : Lm → ˜Lm such that ωm(‘, k) = ˜ωm(ϑ(‘, k)) and

k=1

˜m(˜‘)P

s=1

dβ((‘, k), ϑ(‘, k)) < δ + β

for all (‘, k) ∈ Lm.

Proof. Since dW,∞((L, ω), (˜L, ˜ω)) < δ, there exists µ ∈ M(ω, ˜ω) such that

sup

(a,b)∈R(µ)

d(a, b) < δ

132

where R is the relation associated to the coupling measure µ as in section 2.5.3. Let M0 be
the N × ˜N matrix where mj,k = µ(‘j, ˜‘k) for ‘j ∈ L and ˜‘k ∈ ˜L. An application of Lemma
5.3.1 to M0 yields a matrix M1 such that

• M1 has size N0 × ˜N where N0 is the number of non-zero entries in M0,

• M1 has at most one non-zero entry per row and

• the non-zero entries in each column of M0 and M1 are the same.

Remark 5.3.3 showcases the fact that we can index the entries in M1 as m1
(i,ik),j, where
1 ≤ i ≤ N, 1 ≤ j ≤ m and 1 ≤ k ≤ ni where ni is the number of non-zero elements in the
i-th row of M0. We now apply Lemma 5.3.1 to (M1)T to obtain a matrix M2 such that

• M2 has size ˜N0 × N0 where ˜N0 is the number of non-zero entries in (M1)T ,

• M2 has at most one non-zero entry per row, and

• the non-zero entries in each column of (M1)T and M2 are the same.

Observe that ˜N0 is the number of non-zero entries in M1, and by Remark 5.3.2, it is the
same as the number of non-zero entries in M0. Thus ˜N0 = N0, making M2 a square matrix.
Since the sets of non-zero entries for each column of M2 and (M1)T coincide and M1 has
at most one non-zero entry per row, we conclude that M2 has at most one non-zero entry
in each column and row.

By using the convention in 5.3.3, we can index the elements of M2 as m2

(j,js),(i,ik) with
1 ≤ j ≤ ˜N, 1 ≤ s ≤ ˜nj, where ˜nj is the number of non-zero elements in the j-th column of
M and 1 ≤ i ≤ N, 1 ≤ k ≤ ni where ni is the number of non-zero elements in the i-th row
of M. If we define

{(‘i, k) ∈ M × N : 1 ≤ k ≤ ni}
o

n(˜‘j, s) ∈ M × N : 1 ≤ s ≤ ˜nj

133

N[
˜N[

i=1

Lm =

˜Lm =

j=1

then M := (M2)T defines a measure on (M × N) × (M × N) supported on Lm × ˜Lm. Let

ωm(‘i, k) = X
˜ωm(˜‘j, s) = X

1≤j≤ ˜N

1≤i≤N

X
1≤s≤˜nj
X
1≤k≤nj

m(i,ik),(j,js)

m(i,ik),(j,js).

Notice that the projection on the first component π1 : M × N → M is such that π1(Lm) =
1 (‘i) =

{‘i ∈ M : 1 ≤ i ≤ N} = L and π1(˜Lm) =n˜‘j ∈ M : 1 ≤ j ≤ ˜N
{li} × N, then π−1

o = ˜L. And since π−1
1 (‘i) ∩ Lm = {(‘i, k) ∈ M × N : 1 ≤ k ≤ ni} which implies that
niX

m(i,ik),(j,js)

m(i,ik),(j,js)

{z

entries in the i-th row of M

k=1

1≤k≤ni

ωm(‘i, k) = X
= X
= X

1≤j≤ ˜N

1≤j≤ ˜N

1≤j≤ ˜N

X
X
1≤s≤˜nj

X
 X
1≤s≤˜nj
|

1≤k≤ni

mi,j = ω(‘i).


}

An analogous argument shows that

˜niX

s=1

˜ωm(˜‘i, s) = ˜ω(˜‘j).

Furthermore since M has only one non-zero entry per row and column, it defines a bijection
ϑ between Lm and ˜Lm such that ωm(‘i, k) = ˜ωm(ϑ(‘i, k)). Finally, consider the following
metric on M × N:

dβ((x, s), (y, t)) = d(x, y) + βdδ(s, t).

Using this metric we obtain,

dβ((‘i, k), (˜‘j, s)) = d(‘i, ˜‘j) + βdδ(k, s) < δ + β.

This finishes the proof.

Remark 5.3.5. Index convention for M2 as defined in theorem 5.3.4.

134

i-th block



···
. . .
...
...
...
. . .

...
...

s
...
...

j-th block

···
···
. . .
··· m2
. . .
···

k
···
...

...
···

(j,js),(i,ik)



···
. . .
...
...
...
. . .

···
···
. . .
···
. . .
···

Notation 5.3.6. From this point on we will denote the element (‘j, k) ∈ Lm as ‘j,k ∈ M.
of (M, d). Furthermore d(cid:16)
This notation allows us to make explicit the fact that we will be treating Lm as a multiset

(cid:17) = d(‘j, ‘j) = 0 for any ‘j ∈ L and k, k0 ∈ N.

‘j,k, ‘j,k0

Using the projection π1 in Theorem 5.3.4 we have well defined maps

ι : L → Lm
‘j 7→ ‘j,1

p : Lm → L
‘j,s 7→ π1(lj, s) = ‘j.

Moreover, they induce simplicial maps for the Rips complex at scale α > 0

ι : Rα(L) → Rα(Lm)

and

p : Rα(Lm) → Rα(L).

Definition 5.3.7. Given simplicial maps s, t : K → K0, they are said to be contiguous if
for any simplex σ ∈ K then s(σ) ∪ t(σ) ∈ K0.

Proposition 5.3.8. Let s, t : K → K0 be contiguous simplicial maps between simplicial
complexes K and K0. Then s∗ = t∗ : Hq(K) → Hq(K0) for all q ≥ 0.

Proof. See [51, Theorem 12.5].

Proposition 5.3.9. ιp : Rα(Lm) → Rα(Lm) is contiguous to IRα(Lm) : Rα(Lm) →
Rα(Lm). And pι : Rα(L) → Rα(L) is contiguous to IRα(L) : Rα(L) → Rα(L).

135

Proof. Observe that pι
IRα(L). On the other hand

(cid:16)[‘j1, . . . , ‘jq](cid:17) = p
(cid:16)[‘j1,s1,··· , ‘jq,sq](cid:17) = ι

(cid:16)[‘j1,1, . . . , ‘jq,1](cid:17) = [‘j1, . . . , ‘jq]. Thus, pι =
(cid:16)[‘j1,··· , ‘jq](cid:17) = [‘j1,1,··· , ‘jq,1].

ιp

We want to show that [‘j1,1,··· , ‘jq,1, ‘j1,s1,··· , ‘jq,sq] ∈ Rα(Lm). Since both ‘j1,s1 and
‘j1,1 are copies of ‘j1, we have d(‘j1,s1, ‘j1,1) = 0. Then

diam(cid:16)n

o(cid:17) = diam(cid:16)n
since [‘j1,s1,··· , ‘jq,sq] ∈ Rα(Lm). This finishes the proof.

‘j1,1,··· , ‘jq,1, ‘j1,s1,··· , ‘jq,sq

o(cid:17)

‘j1,s1,··· , ‘jq,sq

< α

Proposition 5.3.9 together with Proposition 5.3.8 implies that ι∗p∗ = IH∗(Rα(Lm)) and
p∗ι∗ = IH∗(Rα(L)), thus H∗(Rα(Lm)) ∼= H∗(Rα(L)) for all α > 0. We end this part with a
Corollary:

Corollary 5.3.10. If Lm is the augmented version of a landmark set L ⊂ M, then their
Rips persistent homology satisfy

bcdR

n (Lm) = bcdR

n (L)

in all dimensions n.

5.3.2 Coordinates old and new

We show next that (L, ω) and (Lm, ωm) create the same coordinates. For brevity, we will
#
i be the cochain map from Ci(Rα(L); R) to
denote by ωjs, the weight ωm(‘j,s). Let p
Ci(Rα(Lm); R) induced by p : Lm → L. Then for a η ∈ C1(Rα(L); R), its image p
1 (η) ∈
#
C1(Rα(Lm); R) is

(cid:16)[‘j, ‘j0](cid:17)

η

0

if j 6= j0
if j = j0,

#
1 (η)(j,s),(j0,t) := p

#
1 (η)([‘j,s, ‘j0,t]) =

p

136

and for a ζ ∈ C0(Rα(L); R), its image p

0 (ζ) ∈ C0(Rα(Lm); R) is
#
(cid:16)[‘j](cid:17)
#
0 (ζ)([‘j,s]) = ζ

#
0 (ζ)(j,s) := p

p

for all 1 ≤ s ≤ Mj and j. If ω is the probability weight on the set L, then the induced
weights ωm on Lm are defined as in Section 5.3.1. Let να be the weight family on the edges
of L. This induces the weight family νm
ωjsωj0t
ωj ωj0 να([‘j, ‘j0])
ωjsωjtρ(α)

if j 6= j0
if j = j0 and s 6= t

α on the edges of Lm defined as

α (‘j,s, ‘j0,t) =
νm



.

These weight families induce inner products on C0(Rα(Lm); R) and C1(Rα(Lm); R) respec-
tively. Recall that both τ = −d+
#
1 (η)) are minimum norm solu-
tions to minimization problems. In particular, they minimize the pairs of objective functions
(F, N) and (Fm, Nm) respectively:

α (η) and τm = −(dm

α )+(p

να([‘j, ‘j0])(cid:16)

ηj,j0 − dαχj,j0

(cid:17)2

F(χ) = X
N(χ) =X

[‘j ,‘

‘j

j0]
ωj(χj)2

and

α ([‘j,s, ‘j0,t])(cid:16)

1 (η)(j,s),(j0,t) − dm
#

α χ(j,s),(j0,t)

p

(cid:17)2

νm

Fm(χ) = X
Nm(χ) = X

[‘j,s,‘

]

j0,t
ωjs(χ(j,s))2

‘j,s

Lemma 5.3.11. For every ζ ∈ C0 (Rα(L); R), the objective functions F(ζ) and N(ζ) are
equal to Fm(p

#
0 (ζ)) respectively.

#
0 (ζ)) and Nm(p

Proof. For any ζ ∈ C0 (Rα(L); R), the objective function Fm is evaluated at its image

137

p

#

Fm(p

0 (ζ) ∈ C0 (Rα(Lm); R) as
#
α ([‘j,s, ‘j0,t])(cid:16)
i νm
 MjX
ωjωj0 να([‘j, ‘j0])(cid:16)
Mj0X

0 (ζ)) = Xh
= X

‘j,s,‘j0,t

ωjsωj0t

s=1

t=1

p

[‘j ,‘j0]

1 (η)(j,s),(j0,t) − (p
#

0 (ζ)(j0,t) − p
#
ηj,j0 − (ζj0 − ζj)(cid:17)2



0 (ζ)(j,s))(cid:17)2

#

where the outermost sum is over all distinct edges [‘j, ‘j0] and the contribution of all terms
of the form p

#
0 (ζ)(j,s) is zero. The above sum becomes

0 (ζ)(j,t) − p
#

#
1 (η)(j,s),(j,t) and p
X
[‘j ,‘j0]

να([‘j, ‘j0])(cid:16)

ηj,j0 − (ζj0 − ζj)(cid:17)2

which is just F(ζ). Similarly,

Nm(p

which is just N(ζ).

0 (ζ)) = X

#

‘j,s

0 (ζ)(j,s))2 = X

#

‘j,s

ωjs(ζj)2 =X

‘j

ωj(ζj)2

ωjs(p

Proposition 5.3.12. The solution τm to the minimization of Fm is just p

#
0 (τ). That is,

(τm)(j,s) := (τm)([‘j,s]) = τ

(cid:16)[‘j](cid:17)

for all j and 1 ≤ s ≤ Mj. Consequently, θm = p

#
1 (η) + (dm

α )(τm) is

(θm)(j,s),(j0,t) := θm([‘j,s, ‘j0,t]) =

(cid:16)[‘j, ‘j0](cid:17)

θ

0

if j 6= j0
if j = j0.

Proof. To show that they are the same, we need to show that the two minimization problems
are equivalent. Recall that

τm =

argmin

χ∈C0(Rα(Lm);R)

X
[‘j,s,‘j0,t]

α ([‘j,s, ‘j0,t])(cid:16)

νm

1 (η)(j,s),(j0,t) − (χ(j0,t) − χ(j,s))(cid:17)2

#

p

.

138

For each χ ∈ C0(Rα(Lm); R), observe that the objective function evaluation
(cid:17)2
1 (η)(j,s),(j0,t) − dm
#
1 (η)(j,s),(j0,t) − (χ(j0,t) − χ(j,s))(cid:17)2
ηj,j0 − (χ(j0,t) − χ(j,s))(cid:17)2

Fm(χ) = X
= X
≥ X

α χ(j,s),(j0,t)

[‘j,s,‘j0,t]

[‘j,s,‘j0,t]

νm

#

p

p



νm

ωjsωj0t

α ([‘j,s, ‘j0,t])(cid:16)
α ([‘j,s, ‘j0,t])(cid:16)
 MjX
Mj0X
να([‘j, ‘j0])(cid:16)
 MjX

να([‘j, ‘j0])

s=1

t=1

ηj,j0

ωjωj0 να([‘j, ‘j0])(cid:16)
(cid:17)2
Mj0X
 MjX

ωjsωj0t
ωjωj0
Mj0X

s=1

t=1

s=1

t=1

να([‘j, ‘j0])2ηj,j0

[‘j ,‘j0]
X
[‘j ,‘j0]
+ X
− X

[‘j ,‘j0]

[‘j ,‘j0]



 .

(χ(j0,t) − χ(j,s))2

ωjsωj0t
ωjωj0

(χ(j0,t) − χ(j,s))

Expanding the square term give us

Now for the second term, we use Lyapunov’s inequality; in particular, that the expected
value of a squared random variable is at least the square of the expected value of the random
variable.

ωjs
ωj

s=1

 MjX
MjX

139

να([‘j, ‘j0])(cid:16)

να([‘j, ‘j0])

ηj,j0

(cid:17)2
 MjX

s=1

j0]

[‘j ,‘

≥ X
+ X
− X

[‘j ,‘

j0]

[‘j ,‘

j0]

χ(j,s) −

j0X

M

t=1

2



ωj0t
ωj0 χ(j0,t)
j0X
ωj0t
ωj0 χ(j0,t)

t=1

M



να([‘j, ‘j0])2ηj,j0

χ(j,s) −

ωjs
ωj

where the third term is a mere rearrangement. Define a 0-cochain ζ ∈ C0 (Rα(L); R) as

ζj := ζ([‘j]) =

ωjs
ωj

χ(j,s).

s=1

Then its image p

0 (ζ) ∈ C0 (Rα(Lm); R) is
#

#
0 (ζ)(j,s) := p

p

0 (ζ)(cid:16)[‘j,s](cid:17) =

#

MjX

s=1

ωjs
ωj

χ(j,s)

for all j and 1 ≤ s ≤ Mj. Therefore, for each χ, we have a found a ζ satisfying

Fm(χ) ≥ Fm(p

0 (ζ)) = F(ζ) ≥ F(τ) = Fm(p
#

#
0 (τ))

where the two equalities follow from lemma 5.3.11. Since p
χ, it is indeed the minimizer of Fm. Furthermore,

#
0 (τ) is independent of choice of

ωjs(χ(j,s))2 =X

ωj

‘j

 MjX

s=1

 .

ωjs
ωj

(χ(j,s))2

‘j,s

Nm(χ) = X
 MjX

ωj

ωjs
ωj

s=1

2

(χ(j,s))

By Lyapunov’s inequality,

Nm(χ) ≥X

‘j

=X

‘j

ωj(ζj)2 = N(ζ) ≥ N(τ) = Nm(p

#
0 (τ))

where ζ is as defined above. Thus, p

#
0 (τ) indeed minimizes Nm as well. Therefore,

τm := (dm

α )+p

#
1 (η) = p

#
0 (τ).

Consequently,

as desired.

(θm)(j,s),(j0,t) = p

#
1 (η)(j,s),(j0,t) + dm

α (τm)(j,s),(j0,t)

#
1 (dα(τ))(j,s),(j0,t)

= ηj,j0 + p
= ηj,j0 + dα(τ)j,j0 = θj,j0

We define a partition of unity {ϕm(j,s)}(j,s) subordinate to the cover

the same fashion the original partition {ϕj}j was defined:

(cid:26)

(cid:27)

in

(j,s)

(‘(j,s))

B α2

ϕm(j,s)(b) =

NP

j=1

ωjsφ(α − d(‘j,s, b))
MjP

ωjsφ(α − d(‘j,s, b))

s=1

.

140

for all j and 1 ≤ s ≤ Mj. Note that if the increasing Lipschitz function used here is the
#
same as the one in ϕj, the right hand side is just ωjs
1 (η), τm, θm,
ωj
and partition of unity {ϕm(j,s)}(j,s), we get circular coordinates:

ϕj(b). With cocycle p

(‘(j,s)) 3 b → exp

B α2

ϕm(r,t)(b)(θm)(j,s),(r,t)

2πi

(τm)(j,s) + X

(r,t)

 .


We conclude in the next theorem that these coordinates agree with the coordinates generated
from the landmark set L.
Theorem 5.3.13. For all b ∈ B α2

(‘j), every j, and every 1 ≤ s ≤ Mj we have that

2πi

(τm)(j,s) + X

(r,t)

exp

ϕm(r,t)(b)(θm)(j,s),(r,t)

(

 = exp


 
τj +X

r

2πi

!)

ϕr(b)θj,r

That is, the circular coordinates induced by the landmark sets L and Lm are equal.

Proof. By Proposition 5.3.12, (τm)(j,s) = τj and (θm)(j,s),(j0,t) = θj,j0. By our choices of
partition of unity, we have that

X

ϕm(r,t)(b)(θm)(j,s),(r,t) =X

MrX

ϕr(b)θj,r =X

r

ωrt
ωr

ϕr(b)θj,r

t=1
for all r and 1 ≤ t ≤ Mr. This finishes the proof.

(r,t)

r

5.3.3 Wasserstein stability

In this section, we combine the previous results to prove Wasserstein stability theorem 5.3.15
for circular coordinates. We begin this section with a lemma relating the Hodge Laplacian
matrix LH to another such matrix defined as:
MiP

˜νm2α−2δ([˜‘i,t, ˜‘j,s])

Lm
H((j, s), (k, t)) =



P
√
ωjsωkt

1
ωjs
− ˜ν2α−2δ([˜‘j,s,˜‘k,t])

1≤i≤N

t=1

if j = k

if j 6= k.

For any matrix A, let e(A) denote the set of distinct eigenvalues.

141

Lemma 5.3.14. Let Lm be the augmented landmark set associated to any L. If LH and Lm
are Hodge Laplacians for C0(Rα(L); R) and C0(Rα(Lm); R) respectively for any α, then

H

e (Lm

H) ⊂ e (LH) .

Proof. Let Λ be an eigenvalue of Lm

H. Then

Lm
HX = ΛX

for some

X =



x11
...

x1M1

...

xN1
...

xN MN



,

(5.5)

where X is an eigenvector for Λ. Therefore, for all 1 ≤ j ≤ N, and 1 ≤ s ≤ Mj, and
i = M1 + ··· + Mj−1 + s,

(Lm

HX)i = (ΛX)i = Λxjs.

(5.6)

By definition of Lm
H and X, we obtain that Λxjs is equal to the terms on the left hand side:
(cid:16)
MkX
MkX

α − d(‘j,s, ‘k,t)(cid:17)
xjs +X

(cid:17)1
2 ρ (α) xjt

− X
+ X

(cid:16)

2 ρ

(cid:17)1
α − d(‘j,s, ‘k,t)(cid:17)

xkt −X

ωjtρ (α) xjs.

ωjsωkt

ωjsωjt

k6=j

t=1

ωktρ

t6=s

(cid:16)

(cid:16)

k6=j

t=1

t6=s

Adding and subtracting the term xjsωjsρ(α), the above can be compactly written as

− NX

MkX

(cid:16)

k=1

t=1

(cid:17)1

2 ρ

(cid:16)

α − d(‘j,s, ‘k,t)(cid:17)

xkt +

ωjsωkt

NX

MkX

k=1

t=1

(cid:16)

α − d(‘j,s, ‘k,t)(cid:17)

xjs

ωktρ

142

. Multiplying this term by

ωj

  ωjs

!1

2

MjX

s=1

ωj

Algebraic rearrangement allows the right hand side to be written as

Λxjs =

+

(cid:16)

2

ωjs

ωj

1
2

(cid:16)

(cid:16)

(cid:16)

xjs

xkt

xkt

t=1

2 ρ

s=1

ωj

s=1

ωj

2

2

2

ρ

2

ρ

ωktρ





k=1

t=1

k=1

t=1

ωjsωkt

!1

2



(cid:18) ωjs
(cid:19)1
2 and adding for all 1 ≤ s ≤ Mj, we obtain
  ωjs
!1
MjX
  ωjs
!1
MjX

α − d(‘j,s, ‘k,t)(cid:17)

  ωkt

− NX
(cid:17)1
MkX
 NX
α − d(‘j,s, ‘k,t)(cid:17)
MkX
MkX
!1
α − d(‘j,s, ‘k,t)(cid:17) (ωk)
(cid:16)


α − d(‘j,s, ‘k,t)(cid:17)MkX
(cid:16)
MkX
α − d(‘j,s, ‘k,t)(cid:17) (ωk)
(cid:16)
α − d(‘j,s, ‘k,t)(cid:17)MkX
 NX
(cid:18) ωjs
(cid:19)1
2 for each j, makes this
α − d(‘j,s, ‘k,t)(cid:17)
(cid:16)
α − d(‘j,s, ‘k,t)(cid:17)

  ωjs
(cid:17)1
 ωjs
!1

 NX

  ωjs
!1
MkP
α − d(‘j,s, ‘k,t)(cid:17) (ωk)
α − d(‘j,s, ‘k,t)(cid:17)
(cid:16)

  ωkt



ωkt = ωk, letting yj :=

1
2 yk +
NX

yj = (LHY)j

MjP
NX



!1

2

yk +



yjρ

(cid:16)

ωkρ

k=1

k=1

xjs

s=1

ρ

k=1

xkt

t=1

ωkt

t=1

ωkt

t=1

t=1

(cid:16)

ωk

ωk

ρ

2

ωk

ωj

1
2

xjs

ωj

ωjs
1
(ωj)
2

xjs

ωj

s=1
This is equal to

MjX
MjX

s=1

−

+

k=1

k=1

 NX
 NX
 MjX
 MjX

s=1

s=1

−

+

MjP

Since

NX

ωjs = ωj and
(cid:16)
(cid:17)1

s=1
− (ωj)
(cid:16)
= − X

k=1

ωjωk

1
2

ρ

2 ρ

k6=j

where Y is the N × 1 vector

This yields

 MjX

s=1

(LHY)j = Λ

(5.7)

 = Λyj = ΛYj

k=1

 .



y1
...

yN

Y =

  ωjs

!1

2

ωj

xjs

143

which implies that Λ is an eigenvalue for LH. Therefore,

e (Lm

H) ⊂ e (LH) .

Now, we prove the Wasserstein stability theorem for circular coordinates.

Theorem 5.3.15. If (L, ω) and (˜L, ˜ω) are two weighted landmarks sets satisfying

dW,∞((L, ω), (˜L, ˜ω)) < δ,

and if h and ˜h are the associated circular valued functions, then
(cid:17) Lρ
(λ2)2 +

 φ(α − δ)
(cid:16)

||h − ˜h||∞ ≤

)δ

(

φ

α − δ − r˜L

min

8πkθk∞
min
j

ωj, min
j

˜ωj

 .

Lφ

φ (α − rL)

Proof. By theorem 5.3.13, we have

|h(b) − ˜h(b)| = |hm(b) − ˜hm(b)|

for each b ∈ M. The new landmark sets (Lm, ωm) and (˜Lm, ˜ωm) are in a weight preserving
bijective correspondence. Hence, by equation 5.4 in the proof of the geometric stability
theorem 5.2.7,

  φ(α − δ)
ψm(b)
where Λ2 is the smallest non-zero eigenvalues of Lm
H. Recall that ||θm||∞ = ||θ||∞, and
observe that ϕm(b) = ϕ(b) and ψm(b) = ψ(b). Moreover, by Lemma 5.3.14, if Λ2 are λ2 are
the smallest non-zero eigenvalues of Lm
H and LH respectively, then Λ2 ≥ λ2. Combining the
above, we obtain

|hm(b) − ˜hm(b)| ≤ 8πkθmk∞δ

(Λ2)2 + Lφ
ϕm(b)

!

Lρ

An argument analogous to the proof of Theorem 5.2.7 yields uniform bounds for all b

|h(b) − ˜h(b)| ≤ 8πkθk∞δ

||h − ˜h||∞ ≤

min

This finishes the proof.

(

8πkθk∞
min
j

ωj, min
j

˜ωj

!

ψ(b)

 φ(α − δ)
 φ(α − δ)
(cid:16)

φ

α − δ − r˜L

Lρ

(λ2)2 + Lφ
ϕ(b)
(cid:17) Lρ
(λ2)2 +

.

)δ



Lφ

φ (α − rL)

144

5.4 Example & Discussion

In this section, we demonstrate circular coordinates through examples. The first example
compares circular coordinates generated from two landmark sets that are close in the Wasser-
stein sense, while the second example is designed to clearly motivate the use of Wasserstein
distance to compare landmarks sets.

Example 5.4.1. We start with a dataset X with 1000 points sample around a unit circle
with Gaussian noise of 0.5. We first select the “blue” landmark set with 40 points by maxmin
sampling. We define the “red” landmark set by choosing the respective nearest neighbours
of the “blue” landmarks. These two landmark sets can be used to create the two associated
circle-valued functions hred and hblue. See Figure 1.4 for the setup and the results. In the
rightmost figure, we plot the pair (hred(b), hblue(b)) for each b ∈ X.The closer the curve is to
the perfect diagonal, the closer the two functions are.

Figure 5.2: An example demonstrating stability of Circular Coordinates: (Left) The data set
X of 1000 points sampled around a circle, with two different sets of 40 landmarks (Middle).
For each of these sets, we obtain a function that assigns to each data point in X a circular
coordinate. (Right) We see a comparison of the two coordinate functions.

Example 5.4.2. To demonstrate the drawbacks of comparing landmarks with Hausdorff
distance, we present a simple example in which landmarks are replicated at a single spot, so
as to have zero Hausdorff distance. Our dataset is generated to lie uniformly around the unit
o16
circle, perturbed by some small Gaussian noise. See Figure 5.3. The first set of landmarks
j=1. The second set of landmarks ˜L is created the

L is selected via maxmin, say L = n

‘j

145

following way:

˜‘j = ‘j for j = 1, ..., 15
˜‘j = ‘16 for j = 16, ..., 15 + n

that is, the final landmark ‘16 is replicated n − 1 times. For each of the landmark sets, we
construct circular coordinates in two fashions to compare them.
In the first case, we simply ignore any notion of weights and calculate the circular coordinates
as originally described in [55].
In the second case, landmarks are weighted so that Wasserstein distance dW,∞((L, ω), (˜L, ˜ω))
is 0. In particular, ωj = 1/16 for every landmark ‘j, while ˜ωj = 1/16 for j = 1, ...15 and
˜ωj = 1/(16n) for i = 16, ..., 15 + n.
In both cases, the circular coordinates built on L are the same, but the coordinates on ˜L
change between the two scenarios. In Fig. 5.3, we see that the first case yields growing error
between the pair of circular coordinates, while in the second case the circular coordinates
remain static regardless of how many replicated landmarks are made.

Figure 5.3: Comparison between Hausdorff and Wasserstein noise: (Left) Example of the
dataset in black and 16 landmarks shown in blue. (Right) Comparison of the number of
replicates against the incurred error between the circular coordinates. Red shows the ini-
tial ‘Zero-Hausdorff-distance’ pair, while green shows the second ‘Zero-Wasserstein-distance’
pair.

Example 5.4.2 demonstrates that Wasserstein distance is the appropriate way to compare

landmarks.

146

BIBLIOGRAPHY

147

BIBLIOGRAPHY

[1] Michał Adamaszek and Henry Adams. The vietoris–rips complexes of a circle. Pacific

Journal of Mathematics, 290(1):1–40, 2017.

[2] John C Baez and James Dolan. Higher-dimensional algebra and topological quantum

field theory. Journal of Mathematical Physics, 36(11):6073–6105, 1995.

[3] Michael Barr and Charles Wells. Category theory for computing science, volume 1.

Prentice Hall New York, 1990.

[4] Ulrich Bauer. Ripser. URL: https://github.com/Ripser/ripser, 2016.
[5] Ulrich Bauer, Michael Kerber, and Jan Reininghaus. Clear and compress: Computing
persistent homology in chunks. In Topological methods in data analysis and visualization
III, pages 103–117. Springer, 2014.

[6] Ulrich Bauer, Michael Kerber, and Jan Reininghaus. Distributed computation of persis-
tent homology. In 2014 proceedings of the sixteenth workshop on algorithm engineering
and experiments (ALENEX), pages 31–38. SIAM, 2014.

[7] Ulrich Bauer and Michael Lesnick.

Induced matchings and the algebraic stability of

persistence barcodes. Journal of Computational Geometry, 6:162–191, 2015.

[8] Hans J. Baues and Ronald Brown. On relative homotogy groups of the product filtration,
the james construction, and a formula of hopf. Journal of pure and applied algebra, 89(1-
2):49–61, 1993.

[9] Jean-Daniel Boissonnat and Clément Maria. Computing persistent homology with var-
ious coefficient fields in a single pass. Journal of Applied and Computational Topology,
3(1-2):59–84, 2019.

[10] Glen E Bredon. Topology and geometry, volume 139. Springer Science & Business

Media, 2013.

[11] Ronald Brown. A new higher homotopy groupoid: the fundamental globular omega-
groupoid of a filtered space. Homology, Homotopy and Applications, 10(1):327–343,
2008.

[12] Peter Bubenik and Nikola Milicevic. Homological algebra for persistence modules. arXiv

preprint arXiv:1905.05744, 2019.

[13] Peter Bubenik and Jonathan A. Scott. Categorification of persistent homology. Discrete

& Computational Geometry, 51(3):600–627, 2014.

[14] Gunnar Carlsson and Benjamin Filippenko. Persistent homology of the sum metric.

Journal of Pure and Applied Algebra, 224(5):106244, 2020.

148

[15] Gunnar Carlsson, Tigran Ishkhanov, Vin De Silva, and Afra Zomorodian. On the local
behavior of spaces of natural images. International journal of computer vision, 76(1):1–
12, 2008.

[16] Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian. Computing multidimensional
persistence. In International Symposium on Algorithms and Computation, pages 730–
739. Springer, 2009.

[17] Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence.

Discrete & Computational Geometry, 42(1):71–93, 2009.

[18] Frédéric Chazal, David Cohen-Steiner, Marc Glisse, Leonidas J Guibas, and Steve Y
In Proceedings of the
Oudot. Proximity of persistence modules and their diagrams.
twenty-fifth annual symposium on Computational geometry, pages 237–246. ACM, 2009.
[19] Frédéric Chazal, Vin De Silva, Marc Glisse, and Steve Oudot. The structure and stability

of persistence modules. Springer, 2016.

[20] Chao Chen and Michael Kerber. Persistent homology computation with a twist. In
Proceedings of the 27th European Workshop on Computational Geometry, volume 11,
pages 197–200, 2011.

[21] René Corbet and Michael Kerber. The representation theorem of persistent homology

revisited and generalized. arXiv preprint arXiv:1707.08864, 2017.

[22] William Crawley-Boevey. Decomposition of pointwise finite-dimensional persistence

modules. Journal of Algebra and its Applications, 14(05):1550066, 2015.

[23] Vin De Silva and Robert Ghrist. Coverage in sensor networks via persistent homology.

Algebraic & Geometric Topology, 7(1):339–358, 2007.

[24] Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Dualities in persistent

(co) homology. Inverse Problems, 27(12):124003, 2011.

[25] Vin De Silva, Dmitriy Morozov, and Mikael Vejdemo-Johansson. Persistent cohomology

and circular coordinates. Discrete & Computational Geometry, 45(4):737–759, 2011.

[26] Alireza Dirafzoon, Alper Bozkurt, and Edgar Lobaton. Geometric learning and topo-
logical inference with biobotic networks. IEEE Transactions on Signal and Information
Processing over Networks, 3(1):200–215, 2016.

[27] Daniel Dugger.

A primer on homotopy colimits.

http://pages.uoregon.edu/ddugger/, 2008.

preprint available at

[28] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence
and simplification. In Proceedings 41st annual symposium on foundations of computer
science, pages 454–463. IEEE, 2000.

[29] RE Edwards. Fourier Series: A Modern Introduction, volume 1. Springer Science &

Business Media, 2012.

149

[30] Samuel Eilenberg and Saunders MacLane. Acyclic models. American journal of math-

ematics, 75(1):189–199, 1953.

[31] Samuel Eilenberg and Norman Steenrod. Foundations of algebraic topology, volume

2193. Princeton University Press, 2015.

[32] Samuel Eilenberg and Joseph A Zilber. On products of complexes. American Journal

of Mathematics, 75(1):200–204, 1953.

[33] Saba Emrani, Thanos Gentimis, and Hamid Krim. Persistent homology of delay em-
IEEE Signal Processing Letters,

beddings and its application to wheeze detection.
21(4):459–463, 2014.

[34] Hitesh Gakhar, Joshua L. Mike, Jose A. Perea, and Luis Polanco. Stability of multiscale

K(G,1) coordinates. In preparation, 2020.

[35] Hitesh Gakhar and Jose A. Perea. Künneth formulae in persistent homology. arXiv

preprint arXiv:1910.05656, 2019.

[36] Hitesh Gakhar and Jose A. Perea. Sliding window embeddings of quasiperiodic func-

tions. In preparation, 2020.

[37] Jerry P Gollub and Harry L Swinney. Onset of turbulence in a rotating fluid. Physical

Review Letters, 35(14):927, 1975.

[38] Loukas Grafakos. Classical fourier analysis, volume 2. Springer, 2008.
[39] Allen Hatcher. Algebraic topology. Cambridge University Press, 2005.
[40] Peter J Hilton and Urs Stammbach. A course in homological algebra, volume 4. Springer

Science & Business Media, 2012.

[41] Morris W Hirsch. Differential topology, volume 33. Springer Science & Business Media,

2012.

[42] Edmund Hlawka, Johannes Schoissengeier, and Rudolf Taschner. Geometric and ana-

lytic number theory. Springer Science & Business Media, 2012.

[43] Dale Husemoller. Fibre bundles, volume 5. Springer, 1966.
[44] Ian Jolliffe. Principal component analysis. Springer, 2011.
[45] Firas A Khasawneh and Elizabeth Munch. Chatter detection in turning using persistent

homology. Mechanical Systems and Signal Processing, 70:527–541, 2016.

[46] Firas A Khasawneh, Elizabeth Munch, and Jose A Perea. Chatter classification in
IFAC-PapersOnLine,

turning using machine learning and topological data analysis.
51(14):195–200, 2018.

[47] Edward N Lorenz. Deterministic nonperiodic flow. Journal of the atmospheric sciences,

20(2):130–141, 1963.

150

[48] Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer

Science & Business Media, 2013.

[49] Facundo Mémoli. Gromov–wasserstein distances and the metric approach to object

matching. Foundations of Computational Mathematics, 11:417–487, 08 2011.

[50] John Milnor. Construction of universal bundles, ii. Annals of Mathematics, pages

430–436, 1956.

[51] James R. Munkres. Elements of Algebraic Topology. Addison Wesley Publishing Com-

pany, 1984.

[52] Steve Y Oudot. Persistence theory: from quiver representations to data analysis, volume

209. American Mathematical Society Providence, RI, 2015.

[53] Jose A. Perea. Persistent homology of toroidal sliding window embeddings. In 2016
IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP),
pages 6435–6439. IEEE, 2016.

[54] Jose A Perea. Multiscale projective coordinates via persistent cohomology of sparse

filtrations. Discrete & Computational Geometry, 59(1):175–225, 2018.

[55] Jose A Perea. Sparse circular coordinates via principal Z-bundles. arXiv preprint

arXiv:1809.09269, 2018.

[56] Jose A. Perea. A brief history of persistence. Morfismos, 23(1):1–16, 2019.
[57] Jose A. Perea. Topological time series analysis. Notices of the American Mathematical

Society, 66(5), 2019.

[58] Jose A Perea, Anastasia Deckard, Steve B Haase, and John Harer. Sw1pers: Sliding
windows and 1-persistence scoring; discovering periodicity in gene expression time series
data. BMC bioinformatics, 16(1):257, 2015.

[59] Jose A. Perea and John Harer. Sliding windows and persistence: An application of
topological methods to signal analysis. Foundations of Computational Mathematics,
15(3):799–838, 2015.

[60] Luis Polanco and Jose A Perea. Coordinatizing data with lens spaces and persistent

cohomology. arXiv preprint arXiv:1905.00350, 2019.

[61] Leonid Polterovich and Egor Shelukhin. Autonomous hamiltonian flows, hofer’s geom-

etry and persistence modules. Selecta Mathematica, 22(1):227–296, 2016.

[62] Leonid Polterovich, Egor Shelukhin, and Vukašin Stojisavljević. Persistence modules
with operators in morse and floer theory. Moscow Mathematical Journal, 17(4):757–786,
2017.

[63] Graeme Segal. Classifying spaces and spectral sequences. Publications Mathématiques

de l’IHÉS, 34:105–112, 1968.

151

[64] Lewis Shilane. Filtered spaces admitting spectral sequence operations. Pacific Journal

of Mathematics, 62(2):569–585, 1976.

[65] David I Spivak. Category theory for the sciences. MIT Press, 2014.
[66] Jeffrey Strom. Modern classical homotopy theory, volume 127. American Mathematical

Society, 2011.

[67] Floris Takens. Detecting strange attractors in turbulence. In Dynamical systems and

turbulence, Warwick 1980, pages 366–381. Springer, 1981.

[68] Christopher J Tralie and Matthew Berger. Topological eulerian synthesis of slow motion
In 2018 25th IEEE International Conference on Image Processing

periodic videos.
(ICIP), pages 3573–3577. IEEE, 2018.

[69] Christopher J Tralie and Jose A Perea. (quasi)-periodicity quantification in video data,

using topology. SIAM Journal on Imaging Sciences, 11(2):1049–1077, 2018.

[70] Michael Usher and Jun Zhang. Persistent homology and floer–novikov theory. Geometry

& Topology, 20(6):3333–3430, 2016.

[71] James W Vick. Homology theory: an introduction to algebraic topology, volume 145.

Springer Science & Business Media, 2012.

[72] Hubert Wagner, Chao Chen, and Erald Vuçini. Efficient computation of persistent
homology for cubical data. In Topological methods in data analysis and visualization II,
pages 91–106. Springer, 2012.

[73] Volkmar Welker, Günter M Ziegler, and Rade T Živaljević. Homotopy colimits–
comparison lemmas for combinatorial applications. Journal für die reine und ange-
wandte Mathematik (Crelles Journal), 1999(509):117–149, 1999.

[74] Inka Wilden, Hanspeter Herzel, Gustav Peters, and Günter Tembrock. Subharmonics,
biphonation, and deterministic chaos in mammal vocalization. Bioacoustics, 9(3):171–
196, 1998.

[75] Boyan Xu, Christopher J Tralie, Alice Antia, Michael Lin, and Jose A Perea. Twisty
takens: A geometric characterization of good observations on dense trajectories. Journal
of Applied and Computational Topology, 3(4):285–313, 2019.

[76] Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete &

Computational Geometry, 33(2):249–274, 2005.

152