Large deviations of extreme eigenvalues of generalized sample covariance matrices

TL;DR

This paper introduces an analytical method to compute the probabilities of rare large deviations in the extreme eigenvalues of generalized sample covariance matrices, advancing understanding of high-dimensional statistical phenomena.

Contribution

The authors develop a new technique that does not require explicit eigenvalue laws, enabling analysis of previously intractable random matrix models and solving open problems in PCA performance.

Findings

Method accurately estimates probabilities of extreme eigenvalue deviations.

Applied to high-dimensional PCA, revealing insights into correlated data.

Effective importance sampling for rare event simulation down to probabilities of 10^{-100}.

Abstract

We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail of the large deviations of the smallest eigenvalue. The technique improves upon past methods by not requiring the explicit law of the eigenvalues, and we apply it to a large class of random matrices that were previously out of reach. In particular, we solve an open problem related to the performance of principal components analysis on highly correlated data, and open the way towards analyzing the high-dimensional landscapes of complex inference models. We probe our results using an importance sampling approach, effectively simulating events with probability as small as $10^{-100}$.