Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions

Loucas Pillaud-Vivien, Francis Bach, Tony Lelièvre, Alessandro Rudi, Gabriel Stoltz
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:2753-2763, 2020.

Abstract

Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré 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 Poincaré constant. As a by-product of the estimation of the Poincaré 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v108-pillaud-vivien20a, title = {Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions}, author = {Pillaud-Vivien, Loucas and Bach, Francis and Lelièvre, Tony and Rudi, Alessandro and Stoltz, Gabriel}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {2753--2763}, year = {2020}, editor = {Chiappa, Silvia and Calandra, Roberto}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/pillaud-vivien20a/pillaud-vivien20a.pdf}, url = {https://proceedings.mlr.press/v108/pillaud-vivien20a.html}, abstract = {Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré 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 Poincaré constant. As a by-product of the estimation of the Poincaré 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.} }
Endnote
%0 Conference Paper %T Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions %A Loucas Pillaud-Vivien %A Francis Bach %A Tony Lelièvre %A Alessandro Rudi %A Gabriel Stoltz %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-pillaud-vivien20a %I PMLR %P 2753--2763 %U https://proceedings.mlr.press/v108/pillaud-vivien20a.html %V 108 %X Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré 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 Poincaré constant. As a by-product of the estimation of the Poincaré 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.
APA
Pillaud-Vivien, L., Bach, F., Lelièvre, T., Rudi, A. & Stoltz, G.. (2020). Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:2753-2763 Available from https://proceedings.mlr.press/v108/pillaud-vivien20a.html.

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