Bregman-Wasserstein divergence: geometry and applications
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| Format: | Preprint |
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2023
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| _version_ | 1866909574316949504 |
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| author | Kainth, Amanjit Singh Rankin, Cale Wong, Ting-Kam Leonard |
| author_facet | Kainth, Amanjit Singh Rankin, Cale Wong, Ting-Kam Leonard |
| contents | The Bregman-Wasserstein divergence is the optimal transport cost when the underlying cost function is given by a Bregman divergence, and arises naturally in fields such as statistics and machine learning. We establish fundamental properties of the Bregman-Wasserstein divergence and propose a novel generalized transport geometry that promotes the Bregman geometry to the space of probability distributions. We provide a probabilistic interpretation involving exponential families and define generalized displacement interpolations compatible with the Bregman geometry. These interpolations are used to derive a generalized Pythagorean inequality, which is of independent interest. Furthermore, we construct a generalized dualistic geometry that lifts the differential geometry of the Bregman divergence to an infinite-dimensional statistical manifold. On the computational side, we demonstrate how Bregman-Wasserstein optimal transport maps can be estimated using neural approaches, establish the well-posedness of Bregman-Wasserstein barycenters, and relate them to Bayesian learning. Finally, we introduce the Bregman-Wasserstein JKO scheme for discretizing Riemannian Wasserstein gradient flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05833 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bregman-Wasserstein divergence: geometry and applications Kainth, Amanjit Singh Rankin, Cale Wong, Ting-Kam Leonard Probability Differential Geometry 53B12 (Primary) 49Q22, 58B20 (Secondary) The Bregman-Wasserstein divergence is the optimal transport cost when the underlying cost function is given by a Bregman divergence, and arises naturally in fields such as statistics and machine learning. We establish fundamental properties of the Bregman-Wasserstein divergence and propose a novel generalized transport geometry that promotes the Bregman geometry to the space of probability distributions. We provide a probabilistic interpretation involving exponential families and define generalized displacement interpolations compatible with the Bregman geometry. These interpolations are used to derive a generalized Pythagorean inequality, which is of independent interest. Furthermore, we construct a generalized dualistic geometry that lifts the differential geometry of the Bregman divergence to an infinite-dimensional statistical manifold. On the computational side, we demonstrate how Bregman-Wasserstein optimal transport maps can be estimated using neural approaches, establish the well-posedness of Bregman-Wasserstein barycenters, and relate them to Bayesian learning. Finally, we introduce the Bregman-Wasserstein JKO scheme for discretizing Riemannian Wasserstein gradient flows. |
| title | Bregman-Wasserstein divergence: geometry and applications |
| topic | Probability Differential Geometry 53B12 (Primary) 49Q22, 58B20 (Secondary) |
| url | https://arxiv.org/abs/2302.05833 |