Large deviations of extreme eigenvalues of generalized sample covariance matrices

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). Copyright (C) 2021 EPLA

Automated summary

This paper presents an analytical technique for computing the probability of rare events in random matrices where the largest eigenvalue is atypically large, and extends to the left tail of the smallest eigenvalue. The new technique does not require explicit knowledge of the eigenvalue law and can be applied to a wider range of random matrices, solving related problems and opening up possibilities for analyzing high-dimensional landscapes of complex inference models. The results are validated using importance sampling to effectively simulate events with extremely small probabilities (down to 10^(-100)).