Asymptotics of Smoothed Wasserstein Distances

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≠ 2 as well as the χ2 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.

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Correspondence to Hong-Bin Chen.

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JNW gratefully acknowledges the support of the Institute for Advanced Study, where a portion of this research was conducted.

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Chen, HB., Niles-Weed, J. Asymptotics of Smoothed Wasserstein Distances. Potential Anal (2021). https://doi.org/10.1007/s11118-020-09895-9

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Keywords

  • Wasserstein distance
  • Heat semigroup
  • Long time asymptotics
  • Euclidean space
  • Polynomial rate

Mathematics Subject Classification 2010

  • 58J65
  • 60J60