Entrywise application of non-linear functions on orthogonally invariant matrices
This addresses theoretical challenges in random matrix theory for researchers, providing insights into non-linear transformations, but it is incremental as it extends known principles to more complex scenarios.
The paper investigates how applying non-linear functions entrywise to symmetric orthogonally invariant random matrices affects their spectral distribution, including multivariate cases with correlations, and finds that a Gaussian equivalence principle holds, meaning the asymptotic effect is equivalent to a linear combination of the matrices plus an independent GOE.
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.