Inference for Projection-Based Wasserstein Distances on Finite Spaces

Okano, Ryo; Imaizumi, Masaaki

doi:10.48550/arxiv.2202.05495

Cited by 1 publication

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“…By bounding the entropy of the function class of Gaussian-smoothed Lipschitz functions and Theorem 1.1 in [34], [35] showed that the function class is P -Donsker under some conditions. [36] explored the limit distribution of projection-based Wasserstein distance based on the same argument of [27] for distributions with finite support. And [37] derived limit law of Sliced p-Wasserstein distance under that P = Q for all p > 1 by Hadamard differentiability theory, but it requires some regularity assumptions such as the absolutely continuity of all one-dimension projections.…”

Section: Introductionmentioning

confidence: 99%

Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance

Xu¹,

Hu²

2022

Preprint

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The Wasserstein distance has been an attractive tool in many fields. But due to its high computational complexity and the phenomenon of the curse of dimensionality in empirical estimation, various extensions of the Wasserstein distance have been proposed to overcome the shortcomings such as the Sliced Wasserstein distance. It enjoys a low computational cost and dimension-free sample complexity, but there are few distributional limit results of it. In this paper, we focus on Sliced 1-Wasserstein distance and its variant max-Sliced 1-Wasserstein distance. We utilize the central limit theorem in Banach space to derive the limit distribution for the Sliced 1-Wasserstein distance. Through viewing the empirical max-Sliced 1-Wasserstein distance as a supremum of an empirical process indexed by some function class, we prove that the function class is P -Donsker under mild moment assumption. Moreover, for computing Sliced p-Wasserstein distance based on Monte Carlo method, we explore that how many random projections that can make sure the error small in high probability. We also provide upper bound of the expected max-Sliced 1-Wasserstein between the true and the empirical probability measures under different conditions and the concentration inequalities for max-Sliced 1-Wasserstein distance are also presented. As applications of the theory, we utilize them for two-sample testing problem.

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