CMX Student/Postdoc Seminar

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the spectral norm. This talk presents our recent results on using semigroup methods to derive sharp nonlinear matrix inequalities. In particular, we show that the classic Bakry—Èmery curvature criterion implies subgaussian concentration for "matrix Lipschitz" functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron—Stein inequalities. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature. We also deduce subexponential matrix concentration from a Poincarè inequality via a short, conceptual argument.