Matrix concentration inequalities for dependent binary random variables

Abstract

We prove Bernstein-type matrix concentration inequalities for linear combinations with matrix coefficients of binary random variables satisfying certain $\ell_\infty$-independence assumptions, complementing recent results by Kaufman, Kyng and Solda. For random variables with the Stochastic Covering Property or Strong Rayleigh Property we prove estimates for general functions satisfying certain direction aware matrix bounded difference inequalities, generalizing and strengthening earlier estimates by the first-named author and Polaczyk. We also demonstrate a general decoupling inequality for a class of Banach-space valued quadratic forms in negatively associated random variables and combine it with the matrix Bernstein inequality to generalize results by Tropp, Chr\'etien and Darses, and Ruetz and Schnass, concerning the operator norm of a random submatrix of a deterministic matrix, drawn by uniform sampling without replacements or rejective sampling, to submatrices given by general Strong Rayleigh sampling schemes.