On the interconnection between the higher-order singular values of real tensors
type: Item
instance of: scholarly article
title: On the interconnection between the higher-order singular values of real tensors
zbMATH Open document ID: 1364.15018
author: Wolfgang Hackbusch
author: André Uschmajew
published in: Numerische Mathematik
publication date: 2017-03-17T00:00:00Z
review text: 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.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 6695820
zbMATH Keywords: matricization
zbMATH Keywords: singular values
zbMATH Keywords: tensor product
describes a project that uses: TensorToolbox
MaRDI profile type: Publication
full work available at URL: s00211-016-0819-9
OpenAlex ID: W2165782098
Wikidata QID: Q42243535
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DOI: 10.1007/S00211-016-0819-9
description: scientific article; zbMATH DE number 6695820