Large Deviations of Extreme Eigenvalues of Generalized Sample Covariance Matrices

08/21/2020

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by   Antoine Maillard, et al.

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Very rare events in which the largest eigenvalue of a random matrix is atypically large have important consequences in statistics, e.g. in principal components analysis, and for studying the rough high-dimensional landscapes encountered in disordered systems in statistical mechanics. These problems lead to consider matrices (1/m) ∑_μ=1^m d_μ𝐳_μ𝐳_μ^†, with {𝐳_μ}_μ=1^m standard Gaussian vectors of size n, and (fixed) real d_μ. In a high-dimensional limit we leverage recent techniques to derive the probability of large deviations of the extreme eigenvalues away from the bulk. We probe our results with Monte-Carlo methods that effectively simulate events with probability as small as 10^-100.