On the Largest Singular Value/Eigenvalue of a Random Tensor

2021/06/14 by Yang, Yuning
#FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2106.07433

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 n₁×⋯× n_d, it is shown that the expectation of its largest singular value is upper bounded by √ n₁+⋯+√ n_d. For the expectation of the largest ℓ^d-singular value, it is upper bounded by 2^(d-1)/(2)∏_j=1^dn_j^(d-2)/(2d)∑^d_j=1n_j^(1)/(2). We also derive the upper bounds of the expectations of the largest Z-/H-(ℓ^d)/M-/C-eigenvalues of symmetric, partially symmetric, and piezoelectric-type Gaussian tensors, which are respectively upper bounded by d√ n, d⋅ 2^(d-1)/(2)n^(d-1)/(2), 2√ m+2√ n, and 3√ n.

Related

• The Edge of Random Tensor Eigenvalues with Deviation
• The expected number of Z-eigenvalues of a real gaussian tensor
• How Many Eigenvalues of a Random Symmetric Tensor are Real?
• Convenient tail bounds for sums of random tensors
• Real eigenvalue/vector distributions of random real antisymmetric tensors