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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.12153 |
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| _version_ | 1866929596405907456 |
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| author | Craig, Katy Yu, Haoqing |
| author_facet | Craig, Katy Yu, Haoqing |
| contents | 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 <= 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12153 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wavelet s-Wasserstein distances for 0 < s <= 1 Craig, Katy Yu, Haoqing Numerical Analysis Optimization and Control 49Q22, 42B35, 65T60 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 <= 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. |
| title | Wavelet s-Wasserstein distances for 0 < s <= 1 |
| topic | Numerical Analysis Optimization and Control 49Q22, 42B35, 65T60 |
| url | https://arxiv.org/abs/2411.12153 |