Abstract
Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $\ell _{2}$ 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.
Highlights
Matrix concentration inequalities describe the probability that a random matrix is close to its expectation, with deviations measured in the 2 operator norm
The purpose of this paper is to advance the theory of semigroups acting on matrixvalued functions and to apply these methods to obtain matrix concentration inequalities for nonlinear random matrix models
We argue that the classical Bakry–Émery curvature criterion for a semigroup acting on real-valued functions ensures that an associated matrix semigroup satisfies a curvature condition
Summary
Matrix concentration inequalities describe the probability that a random matrix is close to its expectation, with deviations measured in the 2 operator norm. The purpose of this paper is to advance the theory of semigroups acting on matrixvalued functions and to apply these methods to obtain matrix concentration inequalities for nonlinear random matrix models. We argue that the classical Bakry–Émery curvature criterion for a semigroup acting on real-valued functions ensures that an associated matrix semigroup satisfies a curvature condition This property further implies local ergodicity of the matrix semigroup, which we can use to prove strong bounds on the trace moments of nonlinear random matrix models. The power of this approach is that the Bakry–Émery condition has already been verified for a large number of semigroups.
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