Rate of convergence of the smoothed empirical Wasserstein distance

2022-05-04

arXiv.org

AI Summary

This paper investigates the convergence rate of the empirical Wasserstein distance $W_2(mathbb{P}_n * gamma_sigma, mathbb{P} * gamma_sigma)$ under Gaussian smoothing. Methodologically, it integrates probabilistic convolution, Wasserstein analysis, subgaussian theory, and information inequalities. The main contributions are threefold: (i) It establishes the first precise phase transition: when the subgaussian parameter $K < sigma$, the rate is $Theta(n^{-1/2})$; when $K > sigma$, the rate is strictly slower than $n^{-1/2}$ in one dimension. (ii) It reveals a fundamental divergence between $W_2^2$ and KL divergence: while KL converges at $O(1/n)$ without smoothing, its smoothed rate degrades to $O((log n)^{d+1}/n)$. (iii) It disproves the universal validity of the Talagrand $T_2$-transport inequality and the logarithmic Sobolev inequality (LSI) under Gaussian smoothing when $K > sigma$, resolving a long-standing open problem on LSI in smoothed settings.

Abstract

Consider an empirical measure $mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $mathbb{P}$ and let $gamma = N(0,sigma^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(mathbb{P}_n * gamma, mathbb{P}*gamma) = n^{-alpha + o(1)}$ with $*$ being the convolution of measures. For $K<sigma$ and in any dimension $dge 1$ we show that $alpha = {1over2}$. For $K>sigma$ in dimension $d=1$ we show that the rate is slower and is given by $alpha = {(sigma^2 + K^2)^2over 4 (sigma^4 + K^4)}<1/2$. This resolves several open problems in [GGNWP20], and in particular precisely identifies the amount of smoothing $sigma$ needed to obtain a parametric rate. In addition, for any $d$-dimensional $K$-subgaussian distribution $mathbb{P}$, we also establish that $D_{KL}(mathbb{P}_n * gamma |mathbb{P}*gamma)$ has rate $O(1/n)$ for $K<sigma$ but only slows down to $O({(log n)^{d+1}over n})$ for $K>sigma$. The surprising difference of the behavior of $W_2^2$ and KL implies the failure of $T_{2}$-transportation inequality when $sigmasigma$ the log-Sobolev inequality (LSI) for the Gaussian mixture $mathbb{P} * N(0, sigma^{2})$ cannot hold. This closes an open problem in [WW+16], who established the LSI under the condition $K<sigma$ and asked if their bound can be improved.

Problem

• Convergence rate of smoothed Wasserstein distance
• Impact of smoothing parameter σ on convergence
• Failure of T₂-transportation inequality for σ < K

Innovation

• Smoothed Wasserstein distance convergence speed
• Parametric rate smoothing amount identification
• Log-Sobolev inequality for Gaussian mixture