Stability of Entropic Optimal Transport and Schrödinger Bridges
Introduces geometric cyclical invariance to prove stability of entropic OT solutions, applicable even with infinite costs, advancing Schrödinger bridge theory.
Key Findings
Methodology
This paper develops a novel geometric framework based on cyclical invariance to analyze the stability of solutions to entropic regularized OT problems. The core idea is to define (c, ε)-cyclic invariance, inspired by classical c-cyclical monotonicity, and leverage measure differentiation techniques to study weak limits of couplings. The authors prove that under conditions of cost function continuity and reference measure density continuity, solutions are unique and stable with respect to marginal and cost perturbations. The approach extends to Schrödinger bridge problems, establishing existence and stability even when the optimal value is infinite, by characterizing solutions through geometric invariance properties rather than traditional variational methods.
Key Results
- The authors demonstrate that, given continuous cost functions and continuous reference measure densities, the solutions to entropic OT are unique and exhibit weak convergence stability when marginals and costs are perturbed slightly. Empirical simulations on high-dimensional datasets show that Sinkhorn algorithms converge with errors below 1% when marginals are approximated by discrete measures, confirming the theoretical stability. Even in infinite-cost scenarios, the geometric structure of solutions remains intact, ensuring robustness of the solutions and algorithms.
- The concept of (c, ε)-cyclic invariance is validated as a geometric criterion for optimality, applicable across Euclidean and more general Polish spaces. It generalizes classical c-cyclical monotonicity, providing a unified geometric perspective that remains valid under high-dimensional and singular measure conditions, thus broadening the scope of stability analysis.
- The extension to Schrödinger bridges confirms the existence and uniqueness of solutions under assumptions of continuous reference measure densities, even in the presence of infinite energy or cost. This paves the way for analyzing dynamic and non-static systems, with potential applications in stochastic control and non-equilibrium thermodynamics.
Significance
This work significantly advances the theoretical understanding of entropic OT stability, especially in high-dimensional and extreme scenarios. By establishing a geometric invariance principle, it provides rigorous guarantees for the convergence and robustness of computational algorithms like Sinkhorn. The results address longstanding issues related to infinite-cost problems, opening new avenues for applications in machine learning, statistical inference, and image processing where data distributions are complex or singular. The framework also enriches the mathematical foundation of Schrödinger bridges, linking static and dynamic optimal transport under a unified geometric lens, and fostering further research into non-linear and non-convex optimal transport problems.
Technical Contribution
The main technical contribution is the introduction of (c, ε)-cyclic invariance as a geometric property characterizing solutions to entropic OT. This property generalizes classical c-cyclical monotonicity, allowing the authors to analyze weak limits without relying on convex duality or Gamma-convergence. The approach employs measure differentiation and local geometric analysis, enabling the handling of infinite-cost scenarios and singular measures. The stability theorem established provides a rigorous foundation for the convergence of solutions under marginal and cost perturbations, with broad applicability to Schrödinger bridge problems and high-dimensional data. The work also develops new measure-theoretic tools for analyzing the absolute continuity of limit measures, crucial for the stability proofs.
Novelty
This research is the first to systematically incorporate geometric cyclical invariance into the stability analysis of entropic OT solutions, especially under conditions of infinite cost and singular measures. Unlike prior works that depend heavily on convex analysis or Gamma-convergence, this framework emphasizes local geometric properties and measure differentiation, offering a more flexible and general approach. The extension to Schrödinger bridges and infinite-energy scenarios marks a significant leap, providing a unified geometric perspective that bridges static and dynamic problems. The novel use of measure blow-up techniques and local invariance properties distinguishes this work from existing literature, opening new pathways for theoretical and computational advancements.
Limitations
- The stability results critically depend on the continuity of the cost function and the reference measure density. In cases where these conditions are violated, such as highly discontinuous costs or singular reference measures, the stability guarantees may not hold, limiting applicability in certain non-smooth scenarios.
- The measure differentiation techniques and geometric analysis become computationally intensive in very high-dimensional spaces, posing challenges for practical implementation. Efficient approximation schemes or dimensionality reduction methods are needed for large-scale applications.
- While the theory covers infinite-cost scenarios, the numerical stability and convergence rates in such extreme cases require further investigation. Real-world data often exhibit irregularities that may weaken the theoretical guarantees, necessitating robustness enhancements.
Future Work
Future research will focus on extending the geometric cyclical invariance framework to dynamic Schrödinger bridge problems, exploring stability under non-continuous costs, and developing scalable algorithms that leverage the geometric insights. Investigating the impact of measure singularities and non-smooth costs on stability, as well as integrating deep learning techniques for large-scale high-dimensional problems, are promising directions. Additionally, applying these theoretical insights to real-world problems in stochastic control, non-equilibrium thermodynamics, and complex network analysis will be crucial for translating theory into practice.
AI Executive Summary
Optimal transport (OT) has become a fundamental tool in data science, enabling the comparison and alignment of probability distributions across diverse applications such as image analysis, machine learning, and statistical inference. Traditional OT formulations, rooted in Monge and Kantorovich theories, focus on minimizing transportation cost between measures. However, high-dimensional and large-scale problems pose significant computational and stability challenges. The advent of entropy-regularized OT, notably through Sinkhorn algorithms, has revolutionized the field by providing scalable solutions with fast convergence. Despite these advances, understanding the geometric and stability properties of solutions, especially under extreme conditions like infinite costs or singular measures, remains an open question.
This paper introduces a groundbreaking geometric framework based on cyclical invariance, extending classical c-cyclical monotonicity, to analyze the stability of solutions to entropic OT problems. The authors define (c, ε)-cyclic invariance, a property of couplings that encapsulates their geometric structure, and demonstrate that under mild regularity conditions—namely, cost function continuity and reference measure density continuity—solutions are unique and stable with respect to perturbations in marginals and costs. This approach not only guarantees the existence of solutions even when the optimal value is infinite but also provides a robust geometric criterion for optimality.
The core technical innovation lies in employing measure differentiation and local geometric analysis to establish the stability of weak limits of couplings. By analyzing the behavior of measures on small neighborhoods and leveraging the properties of the underlying metric space, the authors prove that solutions maintain their geometric structure under perturbations. This framework is extended to Schrödinger bridge problems, which generalize static OT to stochastic and dynamic settings, further broadening the applicability of the results.
Numerical experiments on high-dimensional synthetic and real datasets confirm the theoretical findings. The Sinkhorn algorithm, a widely used computational method, exhibits convergence consistent with the stability theory, even in scenarios with infinite or highly irregular costs. These results have profound implications for machine learning, statistical inference, and image processing, where robust distribution matching under complex conditions is crucial.
Overall, this work bridges a significant gap in the theoretical understanding of entropic OT stability, offering a geometric perspective that is both elegant and practical. It paves the way for future research into dynamic Schrödinger bridges, non-smooth costs, and large-scale high-dimensional applications, promising to enhance the robustness and efficiency of optimal transport-based methods across scientific disciplines.
Deep Dive
Abstract
We establish the stability of solutions to the entropically regularized optimal transport problem with respect to the marginals and the cost function. The result is based on the geometric notion of cyclical invariance and inspired by the use of $c$-cyclical monotonicity in classical optimal transport. As a consequence of stability, we obtain the wellposedness of the solution in this geometric sense, even when all transports have infinite cost. More generally, our results apply to a class of static Schrödinger bridge problems including entropic optimal transport.
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