On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

By Yongxin Chen, Tryphon T. Georgiou, Michele Pavon. Journal of Optimization Theory and Applications, 2014

379 citations recorded by the source corpus.

We take a new look at the relation between the optimal transport problem and the Schrödinger bridge problem from a stochastic control perspective. Our aim is to highlight new connections between the two that are richer and deeper than those previously described in the literature. We begin with an elementary derivation of the Benamou–Brenier fluid dynamic version of the optimal transport problem and provide, in parallel, a new fluid dynamic version of the Schrödinger bridge problem. We observe that the latter establishes an important connection with optimal transport without zero-noise limits and solves a question posed by Eric Carlen in 2006. Indeed, the two variational problems differ by a Fisher information functional. We motivate and consider a generalization of optimal mass transport in the form of a (fluid dynamic) problem of optimal transport with prior. This can be seen as the zero-noise limit of Schrödinger bridges when the prior is any Markovian evolution. We finally specialize to the Gaussian case and derive an explicit computational theory based on matrix Riccati differential equations. A numerical example involving Brownian particles is also provided.

Subfields: Finance, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0b6c88002f9f2c9a4ce990d5fa054acf2f6ab969

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Citations in the corpus

Computational Optimal Transport (2018, methodology): “Dynamical convex formulations for OT over such a cone have been provided Chen et al. [2016b], Jiang et al. [2012]. Some static (Kantorovitch-like) formulations have been also proposed Ning et al. [2015], Peyré et al. [2017], but a mathematically sound theoretical framework is still missing.”
Time reversal of diffusion processes under a finite entropy condition (2021, methodology): “To give some examples, in [4] and [17] a fluid-dynamic (Benamou-Brenier) formulation of entropic optimal transport is derived leveraging time reversal arguments, and similar ideas are also used in [2] in a mean field setting.”
A formula for the time derivative of the entropic cost and applications (2019, background): “The fluid-dynamical formulation of the entropic cost [7, 14, 18, 33] asserts that, at least under (H4), we have”