Plug-in estimation of Schr\"odinger bridges

TL;DR

This paper introduces the Sinkhorn bridge, a novel plug-in estimator for Schr"odinger bridges that leverages static entropic optimal transport to efficiently estimate the dynamic drift without iterative simulations or neural network training.

Contribution

The authors propose a new method for estimating Schr"odinger bridges using static entropic optimal transport potentials, avoiding iterative diffusion simulations and neural network fitting.

Findings

Provably estimates Schr"odinger bridge with convergence depending on target measure's intrinsic dimension.

Does not require iterative forward-backward diffusion simulations.

Combines sampling, entropic optimal transport, and statistical analysis for efficient estimation.

Abstract

We propose a procedure for estimating the Schr\"odinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schr\"odinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.