On the Largest Singular Value/Eigenvalue of a Random Tensor

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 $3\sqrt n$.