# Schr\\"odinger bridge problem via empirical risk minimization Source: [Schr\\"odinger bridge problem via empirical risk minimization]() Collection index: [llms.txt]() Data updated: 2026-02-11T05:00:13.386Z Authors: Denis Belomestny, Alexey Naumov, Nikita Puchkin, Denis Suchkov Published: 2026-02-09T08:12:32.000Z ## Abstract We study the Schr\\"odinger bridge problem when the endpoint distributions are available only through samples. Classical computational approaches estimate Schr\\"odinger 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 Schr\\"odinger 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Intermediate distribution restoration on Single-cell data (intermediate time points ti for i in {1, 2, 3})](): Intermediate distribution restoration; Single-cell data (intermediate time points ti for i in {1, 2, 3}); W1 Score: 0.809. ## Original sources and implementation [Original paper]() [PDF]()