Langevin Diffusion Approximation to Same Marginal Schr\"{o}dinger Bridge

TL;DR

This paper proposes a Langevin diffusion-based approximation to the Schr"{o}dinger bridge, showing its convergence properties and the relationship to the Brenier map as temperature approaches zero.

Contribution

It introduces a novel Langevin diffusion approximation to the Schr"{o}dinger bridge and analyzes its convergence and operator properties at low temperatures.

Findings

The difference between the approximation and the true map is proportional to the score function.

The family of Markov operators admits a derivative at zero temperature given by the Langevin generator.

Operators satisfy an approximate semigroup property at low temperatures.

Abstract

We introduce a novel approximation to the same marginal Schr\"{o}dinger bridge using the Langevin diffusion. As $\varepsilon \downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schr\"{o}dinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $\varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $\mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $\varepsilon > 0$, derived from integrating test functions against the conditional density of the static Schr\"{o}dinger bridge at temperature $\varepsilon$, admits a derivative at $\varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.