On the largest multilinear singular values of higher-order tensors

TL;DR

This paper establishes bounds on the largest multilinear singular values of higher-order tensors, explores the inverse problem for cubic tensors, and connects these results to honeycombs and eigenvalues of Hermitian matrices.

Contribution

It provides new inequalities for multilinear singular values, solves the inverse problem for cubic tensors, and links tensor singular values to eigenvalue problems, offering insights into their joint distribution.

Findings

Derived inequality relating singular values and tensor norm

Proved existence of tensors with prescribed singular values satisfying the inequality

Connected tensor singular values to eigenvalues of Hermitian matrix sums

Abstract

Let $\sigma_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$ \sigma_1^2+\dots+\sigma_{n-1}^2+\sigma_{n+1}^2+\dots+\sigma_{N}^2\leq (N-2)\|\mathcal T\|^2 + \sigma_n^2,\quad n=1,\dots,N, \qquad\qquad (1) $$ where $\|\cdot\|$ denotes the Frobenius norm. We also show that at least for the cubic tensors the inverse problem always has a solution. Namely, for each $\sigma_1,\dots,\sigma_N$ that satisfy (1) and the trivial inequalities $\sigma_1\geq \frac{1}{\sqrt{I}}\|\mathcal T\|,\dots, \sigma_N\geq \frac{1}{\sqrt{I}}\|\mathcal T\|$, there always exists an $I\times \dots\times I$ tensor whose largest multilinear singular values are equal to $\sigma_1,\dots,\sigma_N$. For $N=3$ we show that if the equality $\sigma_1^2+\sigma_2^2= \|\mathcal T\|^2 + \sigma_3^2$ in (1) holds, then $\mathcal T$ is necessarily equal to a sum of multilinear rank-$(L_1,1,L_1)$ and multilinear rank-$(1,L_2,L_2)$ tensors and we give a complete description of all its multilinear singular values. We establish a connection with honeycombs and eigenvalues of the sum of two Hermitian matrices. This seems to give at least a partial explanation of why results on the joint distribution of multilinear singular values are scarce.