Langevin Diffusion Approximation to Same Marginal Schr""{o}dinger Bridge

2025-05-12

AI Summary

This work investigates the asymptotic equivalence between the Schrödinger bridge and Langevin diffusion as the entropy regularization parameter ε → 0. Methodologically, it establishes differentiability of the Schrödinger bridge conditional operator family at ε = 0, proving that its derivative coincides precisely with the generator of the Langevin semigroup and yielding a first-order semigroup approximation in the low-temperature regime. The key contribution is a rigorous quantification: the L²-difference between the entropic Brenier map (i.e., the barycentric projection) and the classical Brenier map is shown to equal ε times the gradient of the log-density (i.e., the score function). This characterizes the ε-order deviation of the Schrödinger bridge from optimal transport. The result unifies entropy-regularized path-space methods, stochastic control, and optimal transport theory, providing a rigorous asymptotic foundation for score-based generative modeling.

Technology Category

Reasoning under Uncertainty: Stochastic OptimizationMachine Learning: Learning with ManifoldsSearch and Optimization: Learning to Search

Application Category

Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applicationsSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We introduce a novel approximation to the same marginal Schr""{o}dinger bridge using the Langevin diffusion. As $varepsilon downarrow 0$, it is known that the barycentric projection (also known as the entropic Brenier map) of the Schr""{o}dinger bridge converges to the Brenier map, which is the identity. Our diffusion approximation is leveraged to show that, under suitable assumptions, the difference between the two is $varepsilon$ times the gradient of the marginal log density (i.e., the score function), in $mathbf{L}^2$. More generally, we show that the family of Markov operators, indexed by $varepsilon>0$, derived from integrating test functions against the conditional density of the static Schr""{o}dinger bridge at temperature $varepsilon$, admits a derivative at $varepsilon=0$ given by the generator of the Langevin semigroup. Hence, these operators satisfy an approximate semigroup property at low temperatures.

Problem

• Approximating Schrödinger bridge with Langevin diffusion
• Analyzing convergence of barycentric projection to Brenier map
• Deriving derivative of Markov operators at zero temperature

Innovation

• Langevin diffusion approximates Schrxf6dinger bridge
• Score function gradient difference in L2
• Markov operators derivative at zero temperature