Optimal Riemannian metric for Poincar\'e inequalities and how to ideally precondition Langevin dynamics

Poincar\'e inequality is a fundamental property that rises naturally in different branches of mathematics. The associated Poincar\'e constant plays a central role in many applications since it governs the convergence of various practical algorithms. For instance, the convergence rate of the Langevin dynamics is exactly given by the Poincar\'e constant. This paper investigates a Riemannian version of Poincar\'e inequality where a positive definite weighting matrix field (\emph{i.e.} a Riemannian metric) is introduced to improve the Poincar\'e constant, and therefore the performances of the associated algorithm. Assuming the underlying measure is a \emph{moment measure}, we show that an optimal metric exists and the resulting Poincar\'e constant is 1. We demonstrate that such optimal metric is necessarily a \emph{Stein kernel}, offering a novel perspective on these complex but central mathematical objects that are hard to obtain in practice. We further discuss how to numerically obtain the optimal metric by deriving an implementable optimization algorithm. The resulting method is illustrated in a few simple but nontrivial examples, where solutions are revealed to be rather sophisticated. We also demonstrate how to design efficient Langevin-based sampling schemes by utilizing the precomputed optimal metric as a preconditioner.