Statistical Estimation of the Poincar{\'e} constant and Application to Sampling Multimodal Distributions

TL;DR

This paper introduces a method to estimate the Poincaré constant from samples, enabling better understanding of convergence rates and identifying key directions in data for improved sampling efficiency.

Contribution

The paper provides a theoretical and experimental framework for estimating the Poincaré constant from data and derives an algorithm to find important low-dimensional directions affecting sampling.

Findings

Successful estimation of Poincaré constants from samples.

Identification of critical directions for sampling and molecular dynamics.

Enhanced sampling techniques leveraging estimated directions.

Abstract

Poincar{\'e} inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincar{\'e} constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincar{\'e} constant. As a by-product of the estimation of the Poincar{\'e} constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.