Nonlinear matrix concentration via semigroup methods
Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the ℓ₂ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical Bakry–Émery 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 due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.
Creative Commons Attribution 4.0 International License. Submitted to EJP on June 30, 2020, final version accepted on December 25, 2020. We thank Ramon van Handel for his feedback on an early version of this manuscript. He is responsible for the observation and proof that matrix Poincaré inequalities are equivalent with scalar Poincaré inequalities, and we are grateful to him for allowing us to incorporate these ideas. We appreciate the thoughtful feedback from the anonymous referees, which has helped us streamline this paper. DH was funded by NSF grants DMS-1907977 and DMS-1912654. JAT gratefully acknowledges funding from ONR awards N00014-17-12146 and N00014-18-12363, and he would like to thank his family for their support in these difficult times.
Submitted - 2006.16562.pdf
Published - euclid.ejp.1610010034.pdf