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Wavelet s-Wasserstein distances for 0 < s <= 1
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 (né Shirdhonkar) and Jacobs showed that, for 0 < s
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| creator | Craig, Katy Yu, Haoqing |
| description | 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 (né Shirdhonkar) and Jacobs showed that, for 0 < s |
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Previous work by Sheory (né 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.</description><identifier>EISSN: 2331-8422</identifier><language>eng</language><publisher>Ithaca: Cornell University Library, arXiv.org</publisher><subject>Equivalence ; Error analysis ; Euclidean geometry ; Euclidean space ; Fourier analysis ; Harmonic analysis ; Translations ; Wavelet analysis</subject><ispartof>arXiv.org, 2024-11</ispartof><rights>2024. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). 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| subjects | Equivalence Error analysis Euclidean geometry Euclidean space Fourier analysis Harmonic analysis Translations Wavelet analysis |
| title | Wavelet s-Wasserstein distances for 0 < s <= 1 |
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