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Register a new userStatistical Estimation of the Poincar{e} constant and Application to Sampling Multimodal Distributions
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Authors
Loucas Pillaud-Vivien
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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.
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The goal of this paper is to push forward the study of those properties of log-concave measures that help to estimate their Poincar{e} constant. First we revisit E. Milmans result [40] on the link between weak (Poincar{e} or concentration) inequalities and Cheegers inequality in the logconcave cases, in particular extending localization ideas and a result of Latala, as well as providing a simpler proof of the nice Poincar{e} (dimensional) bound in the inconditional case. Then we prove alternative transference principle by concentration or using various distances (total variation, Wasserstein). A mollification procedure is also introduced enabling, in the logconcave case, to reduce to the case of the Poincar{e} inequality for the mollified measure. We finally complete the transference section by the comparison of various probability metrics (Fortet-Mourier, bounded-Lipschitz,...).
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We propose that the full Poincar{e} beam with any polarization geometries can be pictorially described by the hybrid-order Poincar{e} sphere whose eigenstates are defined as a fundamental-mode Gaussian beam and a Laguerre-Gauss beam. A robust and efficient Sagnac interferometer is established to generate any desired full Poincar{e} beam on the hybrid-order Poincar{e} sphere, via modulating the incident state of polarization. Our research may provide an alternative way for describing the full Poincar{e} beam and an effective method to manipulate the polarization of light.
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