Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 169, Issue 2, Pages 671-691Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-015-0803-z
Keywords
Optimal transport; Schrodinger bridge; Stochastic control
Funding
- NSF [ECCS-1509387]
- AFOSR [FA9550-12-1-0319, FA9550-15-1-0045]
- University of Padova Research Project CPDA [140897]
- Directorate For Engineering [1665031] Funding Source: National Science Foundation
- Directorate For Engineering
- Div Of Electrical, Commun & Cyber Sys [1509387] Funding Source: National Science Foundation
- Div Of Electrical, Commun & Cyber Sys [1665031] Funding Source: National Science Foundation
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We take a new look at the relation between the optimal transport problem and the Schrodinger 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 Schrodinger 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 Schrodinger 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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