Scalar Poincaré implies matrix Poincaré

Ankit Garg,Tarun Kathuria,Namita Srivastava

doi:10.1214/21-ecp371

Abstract

We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian d×d matrix valued functions, with the same Poincaré constant. This generalizes recent results [ABY19, Kat20] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.

Highlights

There is a long tradition in probability theory of using functional inequalities on a probability space (Ω, Σ, P) to derive concentration inequalities for nice (e.g. Lipschitz) functions f : Ω → R on that space
We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian d × d matrix valued functions, with the same Poincaré constant
The last two of these works in particular studied the notion of matrix Poincaré inequality, in which (1.1) is required to hold for Hd-valued f, E, and Var, with the inequality replaced by the Loëwner ordering on Hd, the space of d × d Hermitian matrices

Summary

Introduction

There is a long tradition in probability theory (see e.g. [GM83, Led99]) of using functional inequalities on a probability space (Ω, Σ, P) to derive concentration inequalities for nice (e.g. Lipschitz) functions f : Ω → R on that space. This generalizes recent results [ABY19, Kat20] establishing such inequalities for specific semigroups and yields new matrix concentration inequalities. The last two of these works in particular (independently) studied the notion of matrix Poincaré inequality, in which (1.1) is required to hold for Hd-valued f , E, and Var, with the inequality replaced by the Loëwner ordering on Hd, the space of d × d Hermitian matrices.

Results

Conclusion