Wavelet s-Wasserstein distances for 0 < s <= 1

TL;DR

This paper introduces a modified wavelet-based s-Wasserstein distance that better captures translation and dilation behaviors of probability measures, improving upon previous formulations and providing a practical embedding into linear spaces.

Contribution

We propose a new wavelet s-Wasserstein distance that corrects previous inaccuracies and better reflects key properties like translation invariance, with demonstrated numerical advantages.

Findings

The new wavelet s-Wasserstein distance is equivalent to the classical W_s distance up to a small error.

Numerical simulations show improved behavior under translations and dilations.

The distance provides a natural linear embedding of the s-Wasserstein space.

Abstract

Motivated by classical harmonic analysis results characterizing H\"older spaces in terms of the decay of their wavelet coefficients, we consider wavelet methods for computing s-Wasserstein type distances. Previous work by Sheory (n\'e Shirdhonkar) and Jacobs showed that, for 0 < s <= 1, the s-Wasserstein distance W_s between certain probability measures on Euclidean space is equivalent to a weighted l_1 difference of their wavelet coefficients. We demonstrate that the original statement of this equivalence is incorrect in a few aspects and, furthermore, fails to capture key properties of the W_s distance, such as its behavior under translations of probability measures. Inspired by this, we consider a variant of the previous wavelet distance formula for which equivalence (up to an arbitrarily small error) does hold for 0 < s < 1. We analyze the properties of this distance, one of which is that it provides a natural embedding of the s-Wasserstein space into a linear space. We conclude with several numerical simulations. Even though our theoretical result merely ensures that the new wavelet s-Wasserstein distance is equivalent to the classical W_s distance (up to an error), our numerical simulations show that the new wavelet distance succeeds in capturing the behavior of the exact W_s distance under translations and dilations of probability measures.