Rate of convergence of the smoothed empirical Wasserstein distance

TL;DR

This paper analyzes the convergence rates of the Gaussian-smoothed empirical Wasserstein distance between the empirical measure $\mathbb{P}_n$ and a true $K$-subgaussian distribution $\mathbb{P}$ after convolution with $\mathcal{N}(0,\sigma^2 I_d)$. It establishes a sharp threshold at $K=\sigma$ separating a parametric $n^{-1/2}$ rate (for $K<\sigma$) from non-parametric rates (in particular, a precise $\alpha$ in 1D when $K>\sigma$). The authors also derive $D_{KL}$ convergence rates, showing $O((\log n)^d/n)$ for all $K$, with a slower polylog penalty when $K>\sigma$, and they connect these findings to failures of $T_2$ transportation and LSI in Gaussian-mixture settings, resolving open problems in the literature. Methodologically, they leverage a χ^2 mutual information framework and Rényi divergences to link information-theoretic quantities to Wasserstein and KL rates, producing both upper and lower bounds that pin down the phase transition and rate constants.

Abstract

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

Theorems & Definitions (49)

• Corollary 1
• Theorem 1
• proof : Proof Idea.
• Proposition 1
• Proposition 2
• Theorem 2
• Remark 1
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• Theorem 3
• Remark 3