Solving Schr\"odinger Bridges via Maximum Likelihood

TL;DR

This paper introduces a novel approach to solving the Schr"odinger bridge problem by establishing an equivalence with maximum likelihood estimation, enabling scalable machine learning techniques for practical applications.

Contribution

It proves an equivalence between SBP and maximum likelihood estimation and proposes a Gaussian process-based numerical method for estimating Schr"odinger bridges.

Findings

Effective numerical procedure demonstrated in simulations

Enables scalable machine learning solutions for SBP

Applicable to dataset alignment and hypothesis testing

Abstract

The Schr\"odinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schr\"odinger bridge remain an active area of research. We prove an equivalence between the SBP and maximum likelihood estimation enabling direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.