Sharp concentration for sums of matrices with Markovian dependence through universality

Abstract

We prove that a sum of random matrices generated by a $\psi$-mixing Markov chain has similar spectral properties to a Gaussian matrix with the same mean and covariance structure. This nonasymptotic universality principle enables sharp concentration inequalities when combined with recent advances in the Gaussian literature. We illustrate the theory with examples, showing how it enables polynomial dimensional improvements relative to previous Markovian matrix concentration results when applied to Wigner-type matrices, and how one can recover sharp limiting values for a model used to study spectral clustering techniques. A key challenge in the proof is that techniques based only on classical cumulants, which can be used when summands are independent, are not sufficient on their own for efficient estimates in a Markovian setting. Our approach exploits Boolean cumulants and a change--of--measure argument.

References in corpus (13)

• Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges
• Random Matrices with Slow Correlation Decay
• Matrix concentration inequalities via the method of exchangeable pairs
• Stability of the Matrix Dyson Equation and Random Matrices with Correlations
• Relations between cumulants in noncommutative probability
• Matrix Concentration Inequalities and Free Probability
• Bernstein type inequality for a class of dependent random matrices
• Concentration Inequalities for Sums of Markov Dependent Random Matrices
• A Matrix Chernoff Bound for Markov Chains and Its Application to Co-occurrence Matrices
• Singular value distribution of dense random matrices with block Markovian dependence
• Estimating the number of clusters of a Block Markov Chain
• Nearly Optimal Latent State Decoding in Block MDPs
• Matrix concentration inequalities for dependent binary random variables