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

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Abstract

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.

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Notes

  1. Over the years, several alternative versions of stochastic mechanics have been proposed by Fényes, Bohm–Vigier, Levy–Krener, Rosenbrock [22–26] to name but a few.

  2. Therefore, W is not a probability measure. Its marginals at each point in time coincide with the Lebesgue measure.

  3. See [30, pp. 7–8] for a justification of employing unbounded path measures in relative entropy problems.

  4. Measure induced by \(\sqrt{\epsilon }w(t)\) on path space \(\varOmega \) with volume measure as initial condition.

  5. This calculation indicates that there may be a limit as \(\epsilon \searrow 0\) of \(\mathrm{inf}\{\epsilon H(Q_\epsilon ,P_\epsilon )\}\) and, hopefully, in suitable sense, of the minimizers. This is indeed the case; see [30, 33, 44] for a precise statement of limiting results.

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Acknowledgments

Research partially supported by the NSF under Grant ECCS-1509387, the AFOSR under Grants FA9550-12-1-0319, and FA9550-15-1-0045 and by the University of Padova Research Project CPDA 140897. Part of the research of M.P. was conducted during a stay at the Courant Institute of Mathematical Sciences of the New York University whose hospitality is gratefully acknowledged.

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  1. University of Minnesota, Minneapolis, MN, USA

    Yongxin Chen & Tryphon T. Georgiou

  2. Università di Padova, Padova, Italy

    Michele Pavon

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  1. Yongxin Chen
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Correspondence to Michele Pavon.

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Chen, Y., Georgiou, T.T. & Pavon, M. On the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint. J Optim Theory Appl 169, 671–691 (2016). https://doi.org/10.1007/s10957-015-0803-z

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