Light Schr\"odinger Bridge

TL;DR

This paper introduces a lightweight, fast, and theoretically justified Schr"odinger Bridge solver that is easy to implement, does not require complex hyperparameter tuning, and can approximate SBs universally, making it practical for moderate-dimensional problems.

Contribution

The paper proposes a novel simple Schr"odinger Bridge solver combining sum-exp quadratic parameterization and energy function interpretation, offering a practical alternative to complex existing methods.

Findings

The solver is simulation-free and runs in minutes on CPU.

It acts as a universal approximator of Schr"odinger Bridges.

The method is inspired by Gaussian mixture models and has theoretical guarantees.

Abstract

Despite the recent advances in the field of computational Schr\"odinger Bridges (SB), most existing SB solvers are still heavy-weighted and require complex optimization of several neural networks. It turns out that there is no principal solver which plays the role of simple-yet-effective baseline for SB just like, e.g., $k$-means method in clustering, logistic regression in classification or Sinkhorn algorithm in discrete optimal transport. We address this issue and propose a novel fast and simple SB solver. Our development is a smart combination of two ideas which recently appeared in the field: (a) parameterization of the Schr\"odinger potentials with sum-exp quadratic functions and (b) viewing the log-Schr\"odinger potentials as the energy functions. We show that combined together these ideas yield a lightweight, simulation-free and theoretically justified SB solver with a simple straightforward optimization objective. As a result, it allows solving SB in moderate dimensions in a matter of minutes on CPU without a painful hyperparameter selection. Our light solver resembles the Gaussian mixture model which is widely used for density estimation. Inspired by this similarity, we also prove an important theoretical result showing that our light solver is a universal approximator of SBs. Furthemore, we conduct the analysis of the generalization error of our light solver. The code for our solver can be found at https://github.com/ngushchin/LightSB