Quantitative Stability of Many-Marginal Schrodinger Bridge

2026-04-16

AI Summary

This work investigates the quantitative stability of multi-marginal Schrödinger bridges under perturbations of the marginal constraints, with particular emphasis on the asymptotic regime where the number of marginals tends to infinity. By analyzing the Kullback–Leibler divergence between bridge measures on path space and leveraging high-order asymptotic expansions of Schrödinger potentials with respect to the regularization parameter, together with Wasserstein-2 geodesic velocity fields and entropic optimal transport theory, the authors establish—for the first time—an upper bound on the stability error that is asymptotically independent of the number of marginals. This bound is governed by the terminal marginal KL divergence and an integrated squared deviation term in time, and it vanishes as the number of marginals increases in the unperturbed setting. Furthermore, asymptotic expansions for the Schrödinger potential and the entropic Brenier map are derived, elucidating the structure of the entropic optimal transport cost and establishing the stability of the Schrödinger functional.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we explore quantitative stability of multi-marginal Schrödinger bridges with respect to the marginal constraints. We focus on the case where the number of marginal constraints is large (i.e. ``many-marginals"). When this number increases, we show that the Kullback--Leibler (KL) divergence between two multi-marginal Schrödinger bridges, as measures on the path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy {that combines} Wasserstein-2 geodesic velocity fields with a log-density gradient term. Our stability upper bound is also asymptotically tight: it converges to zero as the number of marginal constraints increases with unperturbed marginal constraints. To the best of our knowledge, this is the first such stability result that addresses the many-marginal regime, giving error estimates that are asymptotically independent of the number of marginals. To achieve our result, the key step is to derive an asymptotic expansion (of order $k\ge 2$) of Schrödinger potentials with respect to a diminishing regularization coefficient. This result can also be applied to deriving asymptotic expansions of entropic Brenier maps in entropic optimal self-transport problems. As byproducts of our analyses, we also establish the asymptotic expansion of entropic optimal transport cost with respect to the diminishing regularization coefficient when two marginal constraints are sufficiently close. We also prove a stability property of the Schrödinger functional.

Problem

• many-marginal
• Schrödinger bridge
• quantitative stability
• KL divergence
• marginal constraints

Innovation

• many-marginal Schrödinger bridge
• quantitative stability
• asymptotic expansion
• entropic optimal transport
• KL divergence