Large deviations for scaled families of Schr\"odinger bridges with reflection

TL;DR

This paper establishes a large deviation principle for scaled Schr"odinger bridges, extending previous results to more general reference processes, including reflected diffusions, with implications for generative modeling and optimal transport.

Contribution

It extends large deviation results for Schr"odinger bridges to broader reference measures, including reflected diffusions, addressing an open problem and enhancing theoretical understanding.

Findings

Proves large deviation principles for scaled Schr"odinger bridges with general reference measures.

Extends results to reflected diffusions on convex domains.

Provides a theoretical foundation for convergence to optimal transport plans.

Abstract

In this paper, we show a large deviation principle for certain sequences of static Schr\"{o}dinger bridges, typically motivated by a scale-parameter decreasing towards zero, extending existing large deviation results to cover a wider range of reference processes. Our results provide a theoretical foundation for studying convergence of such Schr\"{o}dinger bridges to their limiting optimal transport plans. Within generative modeling, Schr\"{o}dinger bridges, or entropic optimal transport problems, constitute a prominent class of methods, in part because of their computational feasibility in high-dimensional settings. Recently, Bernton et al. established a large deviation principle, in the small-noise limit, for fixed-cost entropic optimal transport problems. In this paper, we address an open problem posed by Bernton et al. and extend their results to hold for Schr\"{o}dinger bridges associated with certain sequences of more general reference measures with enough regularity in a similar small-noise limit. These can be viewed as sequences of entropic optimal transport plans with non-fixed cost functions. Using a detailed analysis of the associated Skorokhod maps and transition densities, we show that the new large deviation results cover Schr\"{o}dinger bridges where the reference process is a reflected diffusion on bounded convex domains, corresponding to recently introduced model choices in the generative modeling literature.