On the largest multilinear singular values of higher-order tensors

Abstract

Let ${\sigma }_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times \cdots \times I_N$ tensor $\mathcal{T}$. We prove that $${\sigma }_1^2+\cdots +{\sigma }_{n-1}^2+{\sigma }_{n+1}^2+\cdots +{\sigma }_N^2\le (N-2)\Vert \mathcal{T}{\Vert }^2+{\sigma }_n^2,n=1,\dots ,N,(1)$$ where $\Vert \cdot \Vert$ 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 ${\sigma }_1\ge \frac{1}{\sqrt{I}}\Vert \mathcal{T}\Vert ,\dots ,{\sigma }_N\ge \frac{1}{\sqrt{I}}\Vert \mathcal{T}\Vert$, there always exists an $I\times \cdots \times I$ tensor whose largest multilinear singular values are equal to ${\sigma }_1,\dots ,{\sigma }_N$. For $N=3$ we show that if the equality ${\sigma }_1^2+{\sigma }_2^2=\Vert \mathcal{T}{\Vert }^2+{\sigma }_3^2$ 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.

19 pages

References in corpus (1)

• Tensor Decomposition for Signal Processing and Machine Learning