Approximation rates of entropic maps in semidiscrete optimal transport
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916491091247104 |
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| author | Sadhu, Ritwik Goldfeld, Ziv Kato, Kengo |
| author_facet | Sadhu, Ritwik Goldfeld, Ziv Kato, Kengo |
| contents | Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, we study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter $\varepsilon$ tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is $O(\sqrt{\varepsilon})$ under the $L^2$-norm with respect to the input measure. In this work, we establish faster, $O(\varepsilon^2)$ rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the $L^2$-norm. For the said dual norm, the $O(\varepsilon^2)$ rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_07947 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation rates of entropic maps in semidiscrete optimal transport Sadhu, Ritwik Goldfeld, Ziv Kato, Kengo Probability Optimization and Control 49Q22, 60F05, 49N15 Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, we study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter $\varepsilon$ tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is $O(\sqrt{\varepsilon})$ under the $L^2$-norm with respect to the input measure. In this work, we establish faster, $O(\varepsilon^2)$ rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the $L^2$-norm. For the said dual norm, the $O(\varepsilon^2)$ rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases. |
| title | Approximation rates of entropic maps in semidiscrete optimal transport |
| topic | Probability Optimization and Control 49Q22, 60F05, 49N15 |
| url | https://arxiv.org/abs/2411.07947 |