Wavelet $$s$$-Wasserstein distances for $$0 < s\leqslant\,1

• Published in: Sampling theory, signal processing, and data analysis Vol. 23; no. 2
• Main Authors: Craig, Katy, Yu, Haoqing
• Format: Journal Article
• Language: English
• Published: 01.12.2025
• ISSN: 2730-5716; 2730-5724
• DOI: 10.1007/s43670-025-00113-4

Motivated by classical harmonic analysis results characterizing Hölder spaces in terms of the decay of their wavelet coefficients, we consider wavelet methods for computing $$s$$ -Wasserstein type distances. Previous work by Sheory (né Shirdhonkar) and Jacobs showed that, for $$0 < s\leqslant1$$ , the $$s$$ -Wasserstein distance $$W_s$$ between certain probability measures on Euclidean space is equivalent to a weighted $$\ell^1$$ 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.