Wavelet -Wasserstein distances for

• 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: Cham Springer International Publishing 01.12.2025
• Subjects: Abstract Harmonic Analysis; Machine Learning; Mathematics; Mathematics and Statistics; Original Article; Signal,Image and Speech Processing; Linearized optimal transport; Wavelets; 42B35; Wasserstein distances; 49Q22; 65T60
• 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 -Wasserstein type distances. Previous work by Sheory (né Shirdhonkar) and Jacobs showed that, for , the -Wasserstein distance between certain probability measures on Euclidean space is equivalent to a weighted 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 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 . We analyze the properties of this distance, one of which is that it provides a natural embedding of the -Wasserstein space into a linear space. We conclude with several numerical simulations. Even though our theoretical result merely ensures that the new wavelet -Wasserstein distance is equivalent to the classical distance (up to an error), our numerical simulations show that the new wavelet distance succeeds in capturing the behavior of the exact distance under translations and dilations of probability measures.