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

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.