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 Title
 Low rank models for multidimensional data recovery and image superresolution
 Creator
 AlQizwini, Mohammed
 Date
 2017
 Collection
 Electronic Theses & Dissertations
 Description

"In the past decade tremendous research efforts focused on signals with specific features, especially sparse and low rank signals. Researchers showed that these signals can be recovered from much smaller number of samples than the Nyquist rate. These efforts were promising for several applications in which the nature of the data is known to be sparse or low rank, but the available samples are much fewer than what is required by the traditional signal processing algorithms to grant an exact...
Show more"In the past decade tremendous research efforts focused on signals with specific features, especially sparse and low rank signals. Researchers showed that these signals can be recovered from much smaller number of samples than the Nyquist rate. These efforts were promising for several applications in which the nature of the data is known to be sparse or low rank, but the available samples are much fewer than what is required by the traditional signal processing algorithms to grant an exact recovery. Our objective in the first part of this thesis is to develop new algorithms for low rank data recovery from few observed samples and for robust low rank and sparse data separation using the Robust Principal Component Analysis (RPCA). Most current approaches in this class of algorithms are based on using the computationally expensive Singular Value Decomposition (SVD) in each iteration to minimize the nuclear norm. In particular, we first develop new algorithms for low rank matrix completion that are more robust to noise and converge faster than the previous algorithms. Furthermore, we generalize our recovery function to the multidimensional tensor domain to target the applications that deal with multidimensional data. Based on this generalized function, we propose a new tensor completion algorithm to recover multidimensional tensors from few observed samples. We also used the same generalized functions for robust tensor recovery to reconstruct the sparse and low rank tensors from the tensor that is formed by the superposition of those parts. The experimental results for this application showed that our algorithms provide comparable performance, or even outperforms, stateoftheart matrix completion, tensor completion and robust tensor recovery algorithms; but at the same time our algorithms converge faster. The main objective of the second part of the thesis develops new algorithms for example based single image superresolution. In this type of applications, we observe a lowresolution image and using some external "example" highresolution  lowresolution images pairs, we recover the underlying highresolution image. The previous efforts in this field either assumed that there is a onetoone mapping between lowresolution and highresolution image patches or they assumed that the highresolution patches span the lower dimensional space. In this thesis, we propose a new algorithm that parts away from these assumptions. Our algorithm uses a subspace similarity measure to find the closes highresolution patch to each lowresolution patch. The experimental results showed that DMCSS achieves clear visual improvements and an average of 1dB improvement in PSNR over stateoftheart algorithms in this field. Under this thesis, we are currently pursuing other low rank and image superresolution applications to improve the performance of our current algorithms and to find other algorithms that can run faster and perform even better."Pages iiiii.
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 Title
 Highdimensional learning from random projections of data through regularization and diversification
 Creator
 Aghagolzadeh, Mohammad
 Date
 2015
 Collection
 Electronic Theses & Dissertations
 Description

Random signal measurement, in the form of random projections of signal vectors, extends the traditional pointwise and periodic schemes for signal sampling. In particular, the wellknown problem of sensing sparse signals from linear measurements, also known as Compressed Sensing (CS), has promoted the utility of random projections. Meanwhile, many signal processing and learning problems that involve parametric estimation do not consist of sparsity constraints in their original forms. With the...
Show moreRandom signal measurement, in the form of random projections of signal vectors, extends the traditional pointwise and periodic schemes for signal sampling. In particular, the wellknown problem of sensing sparse signals from linear measurements, also known as Compressed Sensing (CS), has promoted the utility of random projections. Meanwhile, many signal processing and learning problems that involve parametric estimation do not consist of sparsity constraints in their original forms. With the increasing popularity of random measurements, it is crucial to study the generic estimation performance under the random measurement model. In this thesis, we consider two specific learning problems (named below) and present the following two generic approaches for improving the estimation accuracy: 1) by adding relevant constraints to the parameter vectors and 2) by diversification of the random measurements to achieve fast decaying tail bounds for the empirical risk function.The first problem we consider is Dictionary Learning (DL). Dictionaries are extensions of vector bases that are specifically tailored for sparse signal representation. DL has become increasingly popular for sparse modeling of natural images as well as sound and biological signals, just to name a few. Empirical studies have shown that typical DL algorithms for imaging applications are relatively robust with respect to missing pixels in the training data. However, DL from random projections of data corresponds to an illposed problem and is not wellstudied. Existing efforts are limited to learning structured dictionaries or dictionaries for structured sparse representations to make the problem tractable. The main motivation for considering this problem is to generate an adaptive framework for CS of signals that are not sparse in the signal domain. In fact, this problem has been referred to as 'blind CS' since the optimal basis is subject to estimation during CS recovery. Our initial approach, similar to some of the existing efforts, involves adding structural constraints on the dictionary to incorporate sparse and autoregressive models. More importantly, our results and analysis reveal that DL from random projections of data, in its unconstrained form, can still be accurate given that measurements satisfy the diversity constraints defined later.The second problem that we consider is highdimensional signal classification. Prior efforts have shown that projecting highdimensional and redundant signal vectors onto random lowdimensional subspaces presents an efficient alternative to traditional feature extraction tools such as the principle component analysis. Hence, aside from the CS application, random measurements present an efficient sampling method for learning classifiers, eliminating the need for recording and processing highdimensional signals while most of the recorded data is discarded during feature extraction. We work with the Support Vector Machine (SVM) classifiers that are learned in the highdimensional ambient signal space using random projections of the training data. Our results indicate that the classifier accuracy can be significantly improved by diversification of the random measurements.
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