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Multilinear Algebra in High-Order Data Analysis: Retrieval, Classification and Representation

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posted on 2013-10-24, 00:00 authored by Qun Li
One of the fundamental problems in data analysis is how to represent the data. Real-world signals of practical interest such as color imaging, video sequences and multi-sensor networks, are usually generated by the interaction of multiple factors and thus can be intrinsically represented by higher-order tensors. Application of conventional linear analysis methods to higher-order data tensor representation is typically performed by conversion of the data to very long vectors, thus inevitably losing spatial locality as well as imposing a huge computational and memory burden. As a result, great efforts have been made to extend conventional linear analysis methods that rely on data representation in the form of vectors, for higher-order data analysis. This thesis is dedicated to the study of higher-order data analysis including retrieval, classification and representation, within the mathematical framework provided by multilinear algebra. We first present a higher-order singular value decomposition (HOSVD)-based method for robust indexing and retrieval of higher-order data in responding to various query structures. We prove theoretically that the set of HOSVD unitary matrices of a sub-tensor is equivalent to the corresponding subset of HOSVD unitary matrices of the original tensor. Therefore, if we first arrange all tensors in the database compactly as a higher-order tensor, then we only need to conduct HOSVD once on the total tensor. We then extend linear discriminant analysis (LDA) for higher-order data classification. We propose two multilinear discriminant analysis methods, Direct General Tensor Discriminant Analysis (DGTDA) and Constrained Multilinear Discriminant Analysis (CMDA). Both DGTDA and CMDA seek a tensor-to-tensor projection onto a lower-dimensional tensor subspace, which is most efficient for discrimination. Finally, we propose Generalized Tensor Compressive Sensing (GTCS)--a unified framework for compressive sensing of higher-order tensors. GTCS offers an efficient means for representation of multidimensional data by providing simultaneous acquisition and compression from all tensor modes.



Schonfeld, Dan


Electrical and Computer Engineering

Degree Grantor

University of Illinois at Chicago

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Committee Member

Ansari, Rashid Devroye, Natasha Tuninetti, Daniela Friedland, Shmuel

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