9. Tensor Displacement Structures and Polyspectral Matching
Victor Sorin Grigorascu, Phillip A. Regalia · Society for Industrial and Applied Mathematics eBooks · 1999
9.1 INTRODUCTION This chapter studies the extension of the notion of structured matrices to tensors. These are multi-indexed arrays, in contrast to matrices, which are two-indexed arrays. Such arrays arise while considering higher-order cumulants and the corresponding polyspectra in applications, particularly in blind model identification and approximation problems. While matrices are adequate representations for second-order statistics, higher-order cumulants are more naturally (and more completely) studied in a tensor setting. In this chapter, we examine the displacement rank concept of Ch. 1 for tensors. After a semitutorial presentation of Tucker products and cumulant representations of linear systems, we show links between interpolation of polyspectral values by a linear model and the Tucker factorability of a certain Pick tensor. We also develop a particular higher-order extension of a Schur-type algorithm, based on a novel outer product of tensors. This leads to a pyramidal factorization approach for tensors, which specializes to triangular factorization in the matrix case.