TY - GEN

T1 - Smooth p-Wasserstein Distance

T2 - 38th International Conference on Machine Learning, ICML 2021

AU - Nietert, Sloan

AU - Goldfeld, Ziv

AU - Kato, Kengo

N1 - Publisher Copyright:
Copyright © 2021 by the author(s)

PY - 2021/1/1

Y1 - 2021/1/1

N2 - Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of this framework to high dimensions, we investigate the structural and statistical behavior of the Gaussian-smoothed p-Wasserstein distance Wp(σ), for arbitrary p ≥ 1. After establishing basic metric and topological properties of Wp(σ), we explore the asymptotic statistical behavior of Wp(σ)(µn, µ), where µn is the empirical distribution of n independent observations from µ. We prove that Wp(σ) enjoys a parametric empirical convergence rate of n−1/2, which contrasts the n−1/d rate for unsmoothed Wp when d ≥ 3. Our proof relies on controlling Wp(σ) by a pth-order smooth Sobolev distance d(pσ) and deriving the limit distribution of √nd(pσ)(µn, µ), for all dimensions d. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using Wp(σ), with experiments for p = 2 using a maximum mean discrepancy formulation of d(2σ).

AB - Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of this framework to high dimensions, we investigate the structural and statistical behavior of the Gaussian-smoothed p-Wasserstein distance Wp(σ), for arbitrary p ≥ 1. After establishing basic metric and topological properties of Wp(σ), we explore the asymptotic statistical behavior of Wp(σ)(µn, µ), where µn is the empirical distribution of n independent observations from µ. We prove that Wp(σ) enjoys a parametric empirical convergence rate of n−1/2, which contrasts the n−1/d rate for unsmoothed Wp when d ≥ 3. Our proof relies on controlling Wp(σ) by a pth-order smooth Sobolev distance d(pσ) and deriving the limit distribution of √nd(pσ)(µn, µ), for all dimensions d. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using Wp(σ), with experiments for p = 2 using a maximum mean discrepancy formulation of d(2σ).

UR - http://www.scopus.com/inward/record.url?scp=85161262138&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85161262138

T3 - Proceedings of Machine Learning Research

SP - 8172

EP - 8183

BT - Proceedings of the 38th International Conference on Machine Learning, ICML 2021

PB - ML Research Press

Y2 - 18 July 2021 through 24 July 2021

ER -