Freeness over the diagonal and outliers detection in deformed random
matrices with a variance profile
We study the eigenvalues 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.