The Poincaré inequality and quadratic transportation-variance inequalities

Abstract

It is known that the Poincaré inequality is equivalent to the quadratic transportation-variance inequality (namely Jourdain [10] and most recently Ledoux [12]. We give two alternative proofs to this fact. In particular, we achieve a smaller C V than before, which equals the double of Poincaré constant. Applying the same argument leads to more characterizations of the Poincaré inequality. Our method also yields a by-product as the equivalence between the logarithmic Sobolev inequality and strict contraction of heat flow in Wasserstein space provided that the Bakry-Émery curvature has a lower bound (here the control constants may depend on the curvature bound).Next, we present a comparison inequality between W 2 2 (f µ, µ) and its centralization W 2 2 (fcµ, µ), which may be viewed as some special counterpart of the Rothaus' lemma for relative entropy. Then it yields some new bound of W 2 2 (f µ, µ) associated to the variance of √ f rather than f . As a by-product, we have another proof to derive the quadratic transportationinformation inequality from Lyapunov condition, avoiding the Bobkov-Götze's characterization of the Talagrand's inequality. π∈C(ν,µ) E×E d p (x, y)π(dx, dy) 1/p ,

Analysis of Langevin Monte Carlo from Poincaré to Log-Sobolev

“…Before stating the convergence results, we remark that under (PI), the result of [Liu20] together with standard comparison inequalities imply…”

Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials

“…Using lim q→1 R q (p|π) = H(p|π), convergence in χ 2 divergence implies convergence in KL divergence, which also implies the convergence in total variation by Csiszer-Kullback-Pinsker inequality. In addition, under Poincaré inequality with constant γ, χ 2 divergence also controls the quadratic Wasserstein W 2 distance by W 2 (p, π) 2 ≤ 2 γ χ 2 (p|π) (Liu (2020), Theorem 1.1).…”

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“…which is, by the classical Poincare inequality [26,27], equivalent to the usual norm on X involving the L p − norm of u, as well. In this way, eigenvalue problem (2.1) may be rewritten for λ = 0 and µ = 1 λ , equivalently, as the operator equation…”