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arXiv · 1412.4430

On the relation between optimal transport and Schr\"odinger bridges: A stochastic control viewpoint

Yongxin Chen·Tryphon Georgiou·Michele Pavon
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Abstract

We take a new look at the relation between the optimal transport problem and the Schr\"{o}dinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schr\"{o}dinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schr\"{o}dinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.

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Yongxin Chen, Tryphon Georgiou, Michele Pavon. 2014-12-15. On the relation between optimal transport and Schr\"odinger bridges: A stochastic control viewpoint. https://arxiv.org/abs/1412.4430

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