# Statistical Estimation of the Poincar'e constant and Application to Sampling Multimodal Distributions

arXiv: https://arxiv.org/abs/1910.14564 · HTML: https://mlsanity.com/paper/1910.14564/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Oct 2019.

## Built on by (most cited first, 5 of 5)

- [KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support](https://mlsanity.com/paper/2106.08929/) (2021)
- [Kernelized Diffusion maps](https://mlsanity.com/paper/2302.06757/) (2023)
- [The Impact of Geometric Complexity on Neural Collapse in Transfer Learning](https://mlsanity.com/paper/2405.15706/) (2024)
- [Sharp detection of low-dimensional structure in probability measures via dimensional logarithmic Sobolev inequalities](https://mlsanity.com/paper/2406.13036/) (2024)
- [Operator-Level Quantum Acceleration of Non-Logconcave Sampling](https://mlsanity.com/paper/2505.05301/) (2025)

## Abstract (arXiv)

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.