"With the recent advances in information technology, collection and storage of higher-order datasets such as multidimensional data across multiple modalities or variables have become much easier and cheaper than ever before. Tensors, also known as multiway arrays, provide natural representations for higher-order datasets and provide a way to analyze them by preserving the multilinear relations in these large datasets. These higher-order datasets usually contain large amount of redundant information and summarizing them in a succinct manner is essential for better inference. However, existing data reduction approaches are limited to vector-type data and cannot be applied directly to tensors without vectorizing. Developing more advanced approaches to analyze tensors effectively without corrupting their intrinsic structure is an important challenge facing Big Data applications. This thesis addresses the issue of data reduction for tensors with a particular focus on providing a better understanding of dynamic functional connectivity networks (dFCNs) of the brain. Functional connectivity describes the relationship between spatially separated neuronal groups and analysis of dFCNs plays a key role for interpreting complex brain dynamics in different cognitive and emotional processes. Recently, graph theoretic methods have been used to characterize the brain functionality where bivariate relationships between neuronal populations are represented as graphs or networks. In this thesis, the changes in these networks across time and subjects will be studied through tensor representations. In Chapter 2, we address a multi-graph clustering problem which can be thought as a tensor partitioning problem. We introduce a hierarchical consensus spectral clustering approach to identify the community structure underlying the functional connectivity brain networks across subjects. New information-theoretic criteria are introduced for selecting the optimal community structure. Effectiveness of the proposed algorithms are evaluated through a set of simulations comparing with the existing methods as well as on FCNs across subjects. In Chapter 3, we address the online tensor data reduction problem through a subspace tracking perspective. We introduce a robust low-rank+sparse structure learning algorithm for tensors to separate the low-rank community structure of connectivity networks from sparse outliers. The proposed framework is used to both identify change points, where the low-rank community structure changes significantly, and summarize this community structure within each time interval. Finally, in Chapter 4, we introduce a new multi-scale tensor decomposition technique to efficiently encode nonlinearities due to rotation or translation in tensor type data. In particular, we develop a multi-scale higher-order singular value decomposition (MS-HoSVD) approach where a given tensor is first permuted and then partitioned into several sub-tensors each of which can be represented as a low-rank tensor increasing the efficiency of the representation. We derive a theoretical error bound for the proposed approach as well as provide analysis of memory cost and computational complexity. Performance of the proposed approach is evaluated on both data reduction and classification of various higher-order datasets."--Pages ii-iii.
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