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 n1 × ⋯ × nd, it is shown that the expectation of its largest singular value is upper bounded by n1+⋯+nd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt {{n_1}} + \cdots + \sqrt {{n_d}} $$\end{document}. For the expectation of the largest ℓd-singular value, it is upper bounded by 2d−12∏j=1dnjd−22d∑j=1dnj12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^{{{d - 1} \over 2}}}\prod\nolimits_{j = 1}^d {n_j^{{{d - 2} \over {2d}}}} \sum\nolimits_{j = 1}^d {n_j^{{1 \over 2}}} $$\end{document}. We also derive the upper bounds of the expectations of the largest Z-/H-(ℓd)/M-/C-eigenvalues of symmetric, partially symmetric, and piezoelectric-type Gaussian tensors, which are respectively upper bounded by dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\sqrt n $$\end{document}, d⋅2d−12nd−12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \cdot {2^{{{d - 1} \over 2}}}{n^{{{d - 1} \over 2}}}$$\end{document}, 2m+2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\sqrt m + 2\sqrt n $$\end{document}, and 3n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3\sqrt n $$\end{document}.