When Random Tensors meet Random Matrices

12/23/2021
by   Mohamed El Amine Seddik, et al.
0

Relying on random matrix theory (RMT), this paper studies asymmetric order-d spiked tensor models with Gaussian noise. Using the variational definition of the singular vectors and values of (Lim, 2005), we show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric block-wise random matrix, that is constructed from contractions of the studied tensor with the singular vectors associated to its best rank-1 approximation. Our approach allows the exact characterization of the almost sure asymptotic singular value and alignments of the corresponding singular vectors with the true spike components, when n_i/∑_j=1^d n_j→ c_i∈ [0, 1] with n_i's the tensor dimensions. In contrast to other works that rely mostly on tools from statistical physics to study random tensors, our results rely solely on classical RMT tools such as Stein's lemma. Finally, classical RMT results concerning spiked random matrices are recovered as a particular case.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 23

page 28

05/03/2021

Tubal Matrix Analysis

One of the early ideas started from the 2004 workshop is to regard third...
04/13/2021

Nondegeneracy of eigenvectors and singular vector tuples of tensors

In this article, nondegeneracy of singular vector tuples, Z-eigenvectors...
08/02/2021

A Random Matrix Perspective on Random Tensors

Tensor models play an increasingly prominent role in many fields, notabl...
10/19/2021

Long Random Matrices and Tensor Unfolding

In this paper, we consider the singular values and singular vectors of l...
04/30/2021

Spiked Singular Values and Vectors under Extreme Aspect Ratios

The behavior of the leading singular values and vectors of noisy low-ran...
08/29/2019

Smoothed analysis of the least singular value without inverse Littlewood-Offord theory

We study the lower tail behavior of the least singular value of an n× n ...
12/20/2019

Tensor entropy for uniform hypergraphs

In this paper, we develop a new notion of entropy for uniform hypergraph...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.