Optimal Transport with Tempered Exponential Measures
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911778000076800 |
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| author | Amid, Ehsan Nielsen, Frank Nock, Richard Warmuth, Manfred K. |
| author_facet | Amid, Ehsan Nielsen, Frank Nock, Richard Warmuth, Manfred K. |
| contents | In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, "à-la-Kantorovich", which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, "à-la-Sinkhorn-Cuturi", which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04015 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Transport with Tempered Exponential Measures Amid, Ehsan Nielsen, Frank Nock, Richard Warmuth, Manfred K. Machine Learning Optimization and Control In the field of optimal transport, two prominent subfields face each other: (i) unregularized optimal transport, "à-la-Kantorovich", which leads to extremely sparse plans but with algorithms that scale poorly, and (ii) entropic-regularized optimal transport, "à-la-Sinkhorn-Cuturi", which gets near-linear approximation algorithms but leads to maximally un-sparse plans. In this paper, we show that an extension of the latter to tempered exponential measures, a generalization of exponential families with indirect measure normalization, gets to a very convenient middle ground, with both very fast approximation algorithms and sparsity, which is under control up to sparsity patterns. In addition, our formulation fits naturally in the unbalanced optimal transport problem setting. |
| title | Optimal Transport with Tempered Exponential Measures |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2309.04015 |