# Freeness over the diagonal and outliers detection in deformed random matrices with a variance profile

## TL;DR

This paper analyzes the eigenvalue and singular value distributions of deformed Gaussian matrices with variance profiles, using free probability and deterministic equivalents, to detect outliers and improve low-rank denoising in heteroscedastic noise.

## Contribution

It introduces a fixed point equation approach based on freeness with amalgamation to approximate spectral distributions and identify outliers in deformed random matrices with variance profiles.

## Key findings

- Derived a non-asymptotic fixed point equation for spectral distribution approximation.
- Characterized outlier locations in deformed GUE matrices with variance profiles.
- Demonstrated application to low-rank matrix denoising under heteroscedastic noise.

## Abstract

We study the eigenvalue distribution of a GUE matrix with a variance profile that is perturbed by an additive random matrix that may possess spikes. Our approach is guided by Voiculescu's notion of freeness with amalgamation over the diagonal and by the notion of deterministic equivalent. This allows to derive a fixed point equation to approximate the spectral distribution of certain deformed GUE matrices with a variance profile and to characterize the location of potential outliers in such models in a non-asymptotic setting. We also consider the singular values distribution of a rectangular Gaussian random matrix with a variance profile in a similar setting of additive perturbation. We discuss the application of this approach to the study of low-rank matrix denoising models in the presence of heteroscedastic noise, that is when the amount of variance in the observed data matrix may change from entry to entry. Numerical experiments are used to illustrate our results.