On the interconnection between the higher-order singular values of real tensors
Fix \(d\) \(\geq3\) and positive integers \(n_{1}\geq n_{2}\geq\dots\geq n_{d}\) and consider the tensor product \(\mathbb{R}^{n_{1}}\otimes\mathbb{R}^{n_{2} }\otimes\dots\otimes\mathbb{R}^{n_{d}}\) which is isomorphic to the space \(V\) of all \(n_{1}\times n_{2}\times\dots\times n_{d}\) real arrays.\ For \(X\in V\) we write \(X_{i_{1}i_{2}\dots.i_{d}}\) for the \((i_{1},i_{2},\dots,i_{d})\)-th entry and define \(n_{i}^{c}:=\prod_{j\neq i}n_{j}\). A tensor \(X\in V\) has \(d\) principal matricizations: \(M_{X}^{(j)}\in\mathbb{R}^{n_{j}\times n_{j}^{c}}\) (\(j=1,2,\dots,d\)) where the \(i_{j}\)-th row of matrix \(M_{X}^{(j)}\) consists of all entries \(X_{i_{1}i_{2}\dotsi_{d}}\) with fixed \(i_{j}\) arranged in some fixed ordering with respect to the remaining multi-indices. Let \(\mathbf{\sigma}_{X}^{(j)}\) be the list of \(n_{j}\) singular values of \(M_{X}^{(j)};\) then \(\mathbf{\Sigma}_{X}:=(\mathbf{\sigma}_{X}^{(1)} ,\dots,\mathbf{\sigma}_{X}^{(d)})\) is called the set of higher-order singular values of \(X\). The object of the paper is to investigate the properties of these higher-order singular values. Two principal questions are the following. (Problem 1.4) What can be said about the set of all \(\mathbf{\Sigma}_{X}\) (\(X\in V\))? In particular, given a potential list \(\mathbf{\Sigma}\) of singular values, does there exist a tensor \(X\) such that \(\mathbf{\Sigma} _{X}=\mathbf{\Sigma}\)? (Problem 1.6) Given a tensor \(X\), characterize the tensors \(Y\) for which \(\mathbf{\Sigma}_{X}=\mathbf{\Sigma}_{Y}\). Some progress is made towards solving these problems and a number of conjectures are made, but the details are too complicated to describe here.
scientific article; zbMATH DE number 6695820
scientific article