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 $2^{\frac{d-1}{2}}{\prod }_{j=1}^d{n}_j^{\frac{d-2}{2d}}{\sum }_{j=1}^d{n}_j^{\frac{1}{2}}$. 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 $3\sqrt{n}$.