Deriving Matrix Concentration Inequalities from Kernel Couplings

This paper derives exponential tail bounds and polynomial moment inequalities for the spectral norm deviation of a random matrix from its mean value. The argument depends on a matrix extension of Stein's method of exchangeable pairs for concentration of measure, as introduced by Chatterjee. Recent work of Mackey et al. uses these techniques to analyze random matrices with additive structure, while the enhancements in this paper cover a wider class of matrix-valued random elements. In particular, these ideas lead to a bounded differences inequality that applies to random matrices constructed from weakly dependent random variables. The proofs require novel trace inequalities that may be of independent interest. This paper is based on two independent manuscripts from late 2012 that both used kernel couplings to establish matrix concentration inequalities. One manuscript is by Paulin; the other is by Mackey and Tropp. The authors have combined this research into a unified presentation, with equal contributions from both groups.