Schrödinger bridge problem via empirical risk minimization

TL;DR

This work addresses estimating Schrödinger bridges when endpoint densities are available only through samples by reframing the problem as learning a single positive transformed potential $g$ that satisfies a nonlinear fixed-point equation $g=\mathcal{C}[g]$. The authors replace Sinkhorn iterations with empirical risk minimization over a function class, producing a continuous potential $\widehat{g}_{N,M}$ that can be used with the stochastic-control representation to generate bridge samples. They establish uniform concentration of the empirical risk and derive an approximation-error framework, showing near-parametric rates in the Gaussian-kernel setting via Hermite expansions. Numerically, the ERM-Bridge approach delivers competitive or superior performance to baselines on Swiss-Roll to S-Curve translation, Gaussian mixtures under distribution shift, and single-cell data interpolation, while also offering computational efficiency advantages. This framework advances learning-based Schrödinger bridges by enabling off-sample generalization and flexible function-class representations, with potential impact on data-to-data translation and generative modeling.

Abstract

We study the Schrödinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schrödinger potentials via Sinkhorn iterations on empirical measures and then construct a time-inhomogeneous drift by differentiating a kernel-smoothed dual solution. In contrast, we propose a learning-theoretic route: we rewrite the Schrödinger system in terms of a single positive transformed potential that satisfies a nonlinear fixed-point equation and estimate this potential by empirical risk minimization over a function class. We establish uniform concentration of the empirical risk around its population counterpart under sub-Gaussian assumptions on the reference kernel and terminal density. We plug the learned potential into a stochastic control representation of the bridge to generate samples. We illustrate performance of the suggested approach with numerical experiments.

Figures (3)

• Figure 1: Sample translation from Swiss-Roll to S-Curve and density map for ERM-Bridge at time $t \in [0, 0.25, 0.5, 0.75, 1].$
• Figure 2: Plot of sliced Wasserstein distance as a function of KL between the distribution on the train and sampling for ERM-Bridge and SinkhornBridge.
• Figure 3: Comparison of training and sampling times for ERM-Bridge and SinkhornBridge on the Swiss-Roll to S-Curve translation task. We used 2000 training points for both algorithms.

Theorems & Definitions (25)

• Theorem 3.1
• proof : Sketch of proof
• Theorem 3.2
• proof
• Proposition A.1: Two--sided Gaussian bounds for the fixed point
• Remark A.2
• proof
• Lemma B.1
• proof
• proof : Proof of Theorem \ref{['thm:uniform-concentration-risk']}