[Paper Review] Asymptotics of smoothed Wasserstein distances

Hongbin Chen, Jonathan Niles‐Weed|arXiv (Cornell University)|May 2, 2020

Geometric Analysis and Curvature Flows33 references4 citations

TL;DR

This paper establishes sharp polynomial contraction rates for smoothed Wasserstein distances on $\mathbb{R}^d$ under the heat semigroup (Gaussian convolution), showing that the $2$-Wasserstein distance $W_2^2(\mu*\rho_t, \nu*\rho_t)$ decays as $\Theta(t^{-n})$ when the first $n$ moments of $\mu$ and $\nu$ match but their $(n+1)$-th moments differ, with a positive limiting constant. The result extends to $p$-Wasserstein distances, $\chi^2$ divergence, relative entropy, and total variation, all sharing the same polynomial rate under moment matching.

ABSTRACT

We investigate contraction of the Wasserstein distances on $\mathbb{R}^d$ under Gaussian smoothing. It is well known that the heat semigroup is exponentially contractive with respect to the Wasserstein distances on manifolds of positive curvature; however, on flat Euclidean space---where the heat semigroup corresponds to smoothing the measures by Gaussian convolution---the situation is more subtle. We prove precise asymptotics for the $2$-Wasserstein distance under the action of the Euclidean heat semigroup, and show that, in contrast to the positively curved case, the contraction rate is always polynomial, with exponent depending on the moment sequences of the measures. We establish similar results for the $p$-Wasserstein distances for $p eq 2$ as well as the $χ^2$ divergence, relative entropy, and total variation distance. Together, these results establish the central role of moment matching arguments in the analysis of measures smoothed by Gaussian convolution.

Motivation & Objective

• To resolve the asymptotic behavior of Wasserstein distances under Gaussian smoothing on flat Euclidean space, where exponential contraction fails due to zero curvature.
• To determine under what conditions $W_2(\mu*\rho_t, \nu*\rho_t) \to 0$ as $t \to \infty$ and at what rate.
• To unify the asymptotic behavior of multiple discrepancy measures—Wasserstein, $\chi^2$, relative entropy, and total variation—under Gaussian smoothing.
• To establish that the contraction rate is polynomial, not exponential, and fully determined by moment matching up to order $n$.
• To extend the analysis beyond $p=2$ to general $p$-Wasserstein distances and derive matching asymptotic rates.

Proposed method

• Use of Hermite polynomial expansions and orthogonal decomposition to analyze the difference between smoothed measures $\mu*\rho_t$ and $\nu*\rho_t$.
• Application of the heat semigroup as convolution with the Gaussian kernel $\rho_t(x) = (2\pi t)^{-d/2} e^{-|x|^2/(2t)}$.
• Derivation of asymptotic expansions via Taylor expansion of the density difference up to order $n+1$, relying on moment conditions.
• Use of the $L^2(\mathfrak{g})$ inner product with respect to the standard Gaussian measure $\mathfrak{g}$ to project onto Hermite functions.
• Establishment of uniform Lipschitz bounds on the difference of smoothed densities using estimates on derivatives of remainder terms in the expansion.
• Proof of the key identity $\int H_\alpha(\Theta_t - Q) d\mathfrak{g} = 0$ for $\alpha \in [n+1]$ via moment-matching and orthogonality, leading to the leading-order term in the asymptotics.

Experimental results

Research questions

• RQ1 Under what conditions does $W_2(\mu*\rho_t, \nu*\rho_t)$ converge to zero as $t \to \infty$?
• RQ2 What is the precise rate of decay of $W_2(\mu*\rho_t, \nu*\rho_t)$ when $\mu$ and $\nu$ have matching first $n$ moments but differing $(n+1)$-th moments?
• RQ3 Do other discrepancy measures—such as $\chi^2$ divergence, relative entropy, and total variation—decay at the same polynomial rate as the Wasserstein distance?
• RQ4 How do the asymptotic behaviors of $p$-Wasserstein distances for $p \neq 2$ compare to the $p=2$ case under the same moment-matching condition?
• RQ5 Is the limiting constant in the asymptotic expansion of $t^n W_2^2(\mu*\rho_t, \nu*\rho_t)$ expressible in closed form?

Key findings

• If the first $n$ moments of $\mu$ and $\nu$ match but their $(n+1)$-th moments differ, then $W_2^2(\mu*\rho_t, \nu*\rho_t) = \Theta(t^{-n})$ as $t \to \infty$.
• The rescaled limit $\lim_{t \to \infty} t^n W_2^2(\mu*\rho_t, \nu*\rho_t)$ exists and is a positive constant $c_{\mu,\nu} > 0$, explicitly depending on the $(n+1)$-th moments of $\mu$ and $\nu$.
• The same limiting constant $c_{\mu,\nu}$ governs the asymptotic behavior of the $\chi^2$ divergence and relative entropy between $\mu*\rho_t$ and $\nu*\rho_t$, up to a constant factor.
• For $p$-Wasserstein distances with $p \in [1, \infty)$, the decay rate is $\Theta(t^{-n/2})$, matching the $p=2$ case, with $\liminf_{t \to \infty} t^{n/2} W_p(\mu*\rho_t, \nu*\rho_t) > 0$ and $\limsup < \infty$.
• The total variation distance also decays at rate $\Theta(t^{-n/2})$, but with a different limiting constant than the Wasserstein distances.
• All discrepancy measures analyzed (Wasserstein, $\chi^2$, relative entropy, total variation) exhibit the same polynomial decay rate under the same moment-matching condition, establishing a deep unification in their asymptotic behavior.