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The Data‐Driven Schrödinger Bridge
Author(s) -
Pavon Michele,
Trigila Giulio,
Tabak Esteban G.
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21975
Subject(s) - curse of dimensionality , mathematics , interpolation (computer graphics) , nonlinear system , mathematical optimization , sampling (signal processing) , gaussian , boundary (topology) , importance sampling , computer science , mathematical analysis , artificial intelligence , statistics , monte carlo method , filter (signal processing) , motion (physics) , physics , quantum mechanics , computer vision
Erwin Schrödinger posed—and to a large extent solved—in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet‐IPF‐Sinkhorn type algorithms. Since only samples of the marginals are available, the new approach features constrained maximum likelihood estimation in place of the nonlinear boundary couplings, and importance sampling to propagate the functions ϕ and ϕ ̂ solving the Schrödinger system. This method mitigates the curse of dimensionality, compared to the introduction of grids, which in high dimensions lead to numerically unfeasible methods. The methodology is illustrated in two applications: entropic interpolation of two‐dimensional Gaussian mixtures, and the estimation of integrals through a variation of importance sampling. © 2020 Wiley Periodicals LLC.