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

2018/05/21 by Cosme Louart, Romain Couillet, Louart, Cosme +1 · 3 citations
Mathematics · #60B12 #60B20 #Advanced Combinatorial Mathematics #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1805.08295

openalex publication_date 2018/05/21 · openalex created_date 2019/04/01 · openalex updated_date 2026/07/28

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.

Citations

• On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution
• On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

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