On the Largest Singular Value/Eigenvalue of a Random Tensor

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 $n_1timescdotstimes n_d$, 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 $dsqrt n$, $dcdot 2^{frac{d-1}{2}}n^{frac{d-1}{2}}$, $2sqrt m+2sqrt n$, and $3sqrt n$.