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

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