Concentration of Measure and Large Random Matrices with an application to Sample Covariance Matrices

Abstract

The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential, linear, Lipschitz, convex), a set of elementary tools is laid out that allows for the immediate extension of classical results from random matrix theory involving random concentrated vectors in place of vectors with independent entries. These findings are exemplified here in the context of sample covariance matrices but find a large range of applications in statistical learning and beyond, thanks to the broad adaptability of our hypotheses.

arXiv commentConcentration of the Measure, Random Matrices, Deterministic equivalent, Talagrand Theorem, Hanson-Wright Theorem, Davis Theorem 82 pages, 3 figures

References in corpus (3)

• Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian Mixtures
• On the convergence of the extremal eigenvalues of empirical covariance matrices with dependence
• On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

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