On the largest multilinear singular values of higher-order tensors

Abstract

Let $σ_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$ σ_1^2+\dots+σ_{n-1}^2+σ_{n+1}^2+\dots+σ_{N}^2\leq (N-2)\|\mathcal T\|^2 + σ_n^2,\quad n=1,\dots,N, \qquad\qquad (1) $$ 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 $σ_1,\dots,σ_N$ that satisfy (1) and the trivial inequalities $σ_1\geq \frac{1}{\sqrt{I}}\|\mathcal T\|,\dots, σ_N\geq \frac{1}{\sqrt{I}}\|\mathcal T\|$, there always exists an $I\times \dots\times I$ tensor whose largest multilinear singular values are equal to $σ_1,\dots,σ_N$. For $N=3$ we show that if the equality $σ_1^2+σ_2^2= \|\mathcal T\|^2 + σ_3^2$ 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.