Schrödinger bridge problem via empirical risk minimization

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.

💡 Research Summary

The paper tackles the Schrödinger bridge problem (SBP) in the realistic setting where the two marginal distributions are only available through finite samples. Classical computational pipelines first solve an entropy‑regularized optimal transport (EOT) problem on the empirical measures using the Sinkhorn algorithm, obtain discrete approximations of the Schrödinger potentials (ν₀, ν_T), and then smooth and differentiate these discrete potentials to construct a time‑inhomogeneous drift for the controlled diffusion that realizes the bridge. This approach suffers from two major drawbacks: (1) the potentials are defined only at the sampled points and must be interpolated off‑sample, which can be unstable; (2) the overall error mixes optimization error (finite Sinkhorn iterations), statistical error (finite samples), and smoothing/interpolation error, making rigorous generalization analysis difficult.

The authors propose a fundamentally different route: they rewrite the Schrödinger system as a single nonlinear fixed‑point equation for a transformed positive function
 g(y) = ρ_T(y) ν_T(y).
Defining D_g(x) = ∫ q_T(x,z) ρ_T(z) g(z) dz and the integral operator
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