Tensor decompositions for signal processing applications: from two-way to multiway component analysis

Abstract

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile multi-way data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under very mild and natural conditions. Benefiting from the power of multilinear algebra as their mathematical backbone, data analysis techniques using tensor decompositions are shown to find more general latent components in the data than matrix-based methods, and to have great flexibility in the choice of constraints that match data properties. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We also cover computational aspects and show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the blessings of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these blessings also extend to vector/matrix data through tensorization.