Schrödinger bridge problem via empirical risk minimization

• URL: http://arxiv.org/abs/2602.08374v1
• Date: Mon, 09 Feb 2026 08:12:32 GMT
• Title: Schrödinger bridge problem via empirical risk minimization
• Authors: Denis Belomestny, Alexey Naumov, Nikita Puchkin, Denis Suchkov,
• Abstract summary: We study the Schrdinger bridge problem when the endpoint distributions are available only through samples.<n>We rewrite the Schrdinger system in terms of a single positive transformed potential that satisfies a nonlinear fixed-point equation.<n>We plug the learned potential into a control representation of the bridge to generate samples.
• Score: 12.467558505686588
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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