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.
