Entrywise application of non-linear functions on orthogonally invariant matrices

TL;DR

This paper studies how applying non-linear functions entrywise to orthogonally invariant matrices affects their spectral distribution, revealing a Gaussian equivalence principle that simplifies the asymptotic behavior.

Contribution

It introduces a Gaussian equivalence principle for the entrywise application of non-linear functions on orthogonally invariant matrices, including multivariate cases with correlations.

Findings

Gaussian equivalence principle holds asymptotically

ReLU and max functions exemplify the theory

Asymptotic spectral effects are linear combinations plus GOE

Abstract

In this article, we investigate how the entrywise application of a non-linear function to symmetric orthogonally invariant random matrix ensembles alters the spectral distribution. We treat also the multivariate case where we apply multivariate functions to entries of several orthogonally invariant matrices; where even correlations between the matrices are allowed. We find that in all those cases a Gaussian equivalence principle holds, that is, the asymptotic effect of the non-linear function is the same as taking a linear combination of the involved matrices and an additional independent GOE. The ReLU-function in the case of one matrix and the max-function in the case of two matrices provide illustrative examples.