On the Largest Singular Value/Eigenvalue of a Random Tensor

TL;DR

This paper derives non-asymptotic upper bounds for the expected largest singular values and eigenvalues of various random tensors, including Gaussian tensors and symmetric types, highlighting their dependence on tensor dimensions.

Contribution

It provides new non-asymptotic upper bounds for the expectations of the largest singular values and eigenvalues of different classes of random tensors, extending previous results.

Findings

Expectation of largest singular value bounded by sum of square roots of dimensions

Expectation of largest $ ext{ell}^d$-singular value bounded by a product and sum involving tensor dimensions

Upper bounds for eigenvalues of symmetric and structured Gaussian tensors

Abstract

This short note presents upper bounds of the expectations of the largest singular values/eigenvalues of various types of random tensors in the non-asymptotic sense. For a standard Gaussian tensor of size $n_1\times\cdots\times n_d$, it is shown that the expectation of its largest singular value is upper bounded by $\sqrt {n_1}+\cdots+\sqrt {n_d}$. For the expectation of the largest $\ell^d$-singular value, it is upper bounded by $2^{\frac{d-1}{2}}\prod_{j=1}^{d}n_j^{\frac{d-2}{2d}}\sum^d_{j=1}n_j^{\frac{1}{2}}$. We also derive the upper bounds of the expectations of the largest Z-/H-($\ell^d$)/M-/C-eigenvalues of symmetric, partially symmetric, and piezoelectric-type Gaussian tensors, which are respectively upper bounded by $d\sqrt n$, $d\cdot 2^{\frac{d-1}{2}}n^{\frac{d-1}{2}}$, $2\sqrt m+2\sqrt n$, and $3\sqrt n$.