The Data‐Driven Schrödinger Bridge

Abstract

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 trueϕ̂ 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.

Solving Schrödinger Bridges via Maximum Likelihood

“…This is formalised in Theorem 1 and Observation 1. This differs to prior approaches such as [ 10 ] where their maximum likelihood formulation solves a density estimation problem. This allows the application of many regression methods from machine learning that scale well to high dimensional problems.…”

“…While we do not directly use the stochastic control formulations, the drift-based formulation serves as inspiration for our iterative scheme. Specifically, the existence and parametrisation of an optimal drift as shown in [ 10 , 20 ].…”

“…Note that this formulation can be parametrised with any method of choice. We chose GPs; however, neural networks would also be well suited, We solve the aforementioned regression objectives using GPs [ 11 ] motivated by the connection between the drift of stochastic differential equations and GPs [ 12 ], We provide a conceptual comparison with the approach by [ 10 ] and detail why the density estimation formulation in [ 10 ] scales poorly with dimension, Finally we re-implement the approach by [ 10 ] and compare approaches across a series of numerical experiments. Furthermore we empirically show how our approach works well in dimensions higher than 2.…”

“…We provide a conceptual comparison with the approach by [ 10 ] and detail why the density estimation formulation in [ 10 ] scales poorly with dimension,…”

“…Finally we re-implement the approach by [ 10 ] and compare approaches across a series of numerical experiments. Furthermore we empirically show how our approach works well in dimensions higher than 2.…”