Matrix Poincaré inequalities and concentration

Abstract

We show that any probability measure satisfying a Matrix Poincaré inequality with respect to some reversible Markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carré du champ operator. This extends to the matrix setting a classical phenomenon in the scalar case. Moreover, the proof gives rise to new matrix trace inequalities which could be of independent interest. We then apply this general fact by establishing matrix Poincaré inequalities to derive matrix concentration inequalities for Gaussian measures, product measures and for Strong Rayleigh measures. The latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables.

Scalar Poincaré Implies Matrix Poincaré

“…They showed that a matrix Poincaré inequality generically implies concentration bounds in the operator norm similar to those in the scalar case 1 . They then proceeded to prove matrix Poincaré inequalities for several interesting classes of measures (product [ABY19], Gaussian [ABY19], Strongly Rayleigh [ABY19, Kat19]) on a case by case basis, often mimicking the scalar proofs but requiring significant additional work to handle the noncommutativity of matrices.…”

“…Theorem 1.1 implies that any reversible Markov semigroup satisfying a Poincaré inequality satisfies an exponential matrix concentration inequality; in particular [ABY19, Theorem 1.1] holds with the "matrix Poincaré" assumption replaced by "Poincaré". It also allows us to deduce all of the matrix Poincaré inequalities derived in [ABY19,Kat19] from their known scalar counterparts, and yields new matrix Poincaré and concentration inequalities, notably for Completely Log Concave (i.e., Lorentzian [BH19]) measures via [ALGV19, Theorem 1.1].…”

“…[ABY19] also used the related notion of matrix carré du champ operator and obtained more refined bounds than[Kat19] in terms of it.…”