Building the Bridge of Schr\"odinger: A Continuous Entropic Optimal Transport Benchmark

TL;DR

This paper introduces a new benchmark for continuous entropic optimal transport and Schr"odinger Bridge problems, enabling evaluation of neural solvers in high-dimensional spaces like images.

Contribution

It proposes a generic method to generate probability distribution pairs with known solutions, facilitating the assessment of neural EOT/SB solvers.

Findings

Benchmark pairs reveal solver accuracy in high-dimensional spaces

Method applies to various OT formulations including EOT and SB

Code available for reproducibility and further testing

Abstract

Over the last several years, there has been significant progress in developing neural solvers for the Schr\"odinger Bridge (SB) problem and applying them to generative modelling. This new research field is justifiably fruitful as it is interconnected with the practically well-performing diffusion models and theoretically grounded entropic optimal transport (EOT). Still, the area lacks non-trivial tests allowing a researcher to understand how well the methods solve SB or its equivalent continuous EOT problem. We fill this gap and propose a novel way to create pairs of probability distributions for which the ground truth OT solution is known by the construction. Our methodology is generic and works for a wide range of OT formulations, in particular, it covers the EOT which is equivalent to SB (the main interest of our study). This development allows us to create continuous benchmark distributions with the known EOT and SB solutions on high-dimensional spaces such as spaces of images. As an illustration, we use these benchmark pairs to test how well existing neural EOT/SB solvers actually compute the EOT solution. Our code for constructing benchmark pairs under different setups is available at: https://github.com/ngushchin/EntropicOTBenchmark.