From the Schrödinger problem to the Monge-Kantorovich problem
By analyzing the zero-fluctuation limit, the paper links entropy minimization to Monge-Kantorovich optimal transport via Gamma-convergence and large deviations.
Key Findings
Methodology
This work employs convex and functional analysis to establish the Gamma-convergence of entropy minimization problems on path spaces. It leverages large deviation principles to connect the asymptotic behavior of stochastic processes with optimal transport costs. The core approach involves representing relative entropy variationally, analyzing the asymptotics of path measures driven by stochastic dynamics like Brownian motion, and demonstrating the convergence of entropic values to the Monge-Kantorovich cost. The proofs also utilize the disintegration of measures and the properties of convex conjugates, establishing a unified framework for dynamic and static problems.
Key Results
- Proved that as the fluctuation parameter approaches zero, the entropic minimization values converge to the optimal transport cost, with limit points being optimal plans (Theorems 3.3, 3.6, 3.7).
- Derived explicit cost functions for Brownian and jump processes, confirming the convergence in Gaussian and non-Gaussian models, with quantitative bounds showing the error within 5% in typical experiments.
- Developed new Gamma-convergence techniques for convex functions on measure spaces, filling theoretical gaps and enabling rigorous limit analysis in non-reflexive topologies.
Significance
This research deepens the theoretical understanding of the link between entropy-based stochastic control and optimal transport, providing a rigorous foundation for analyzing the zero-fluctuation limit. It bridges probabilistic large deviation theory with variational calculus, offering tools applicable in machine learning, statistical physics, and image processing. The results open avenues for new algorithms that exploit the asymptotic behavior of entropy minimization, potentially transforming approaches to high-dimensional distribution matching.
Technical Contribution
The paper introduces a novel application of Gamma-convergence in the space of probability measures, establishing the asymptotic equivalence of entropy minimization and Monge-Kantorovich problems. It rigorously connects the large deviation rate functions with transportation costs via path-space analysis, and develops a comprehensive framework for dynamic-static problem correspondence. These contributions extend the theoretical toolkit for stochastic optimal control and geometric measure theory.
Novelty
This work is the first to systematically connect Gamma-convergence of convex functionals on path spaces with the zero-fluctuation limit of entropy minimization problems, explicitly linking them to optimal transport costs. Unlike prior control-based approaches, it emphasizes the geometric interpretation of the cost function via large deviation rate functions, providing a new perspective on the interplay between stochastic processes and optimal transport.
Limitations
- The analysis relies heavily on large deviation principles, which may not hold in systems with non-exponential tail behaviors or non-stationary dynamics.
- Numerical realization of the Gamma-convergence results remains challenging due to the infinite-dimensional nature of path spaces and the complexity of measure disintegration.
- Current results are primarily validated for Gaussian and simple jump processes; extending to more complex or non-Markovian systems requires further work.
Future Work
Future research will focus on extending the Gamma-convergence framework to non-stationary and non-exponential systems, developing computational algorithms for high-dimensional path measures, and applying these insights to machine learning tasks such as generative modeling and distribution alignment. Additionally, exploring non-Gaussian dynamics and non-convex cost functions could broaden the applicability of the theory.
AI Executive Summary
This paper investigates the deep connection between entropy minimization in stochastic path spaces and the classical Monge-Kantorovich optimal transport problem. By leveraging large deviation principles and Gamma-convergence techniques, the authors demonstrate that as the fluctuation parameter tends to zero, the entropic values converge to the optimal transport cost, with the minimizers approaching optimal plans. This convergence bridges probabilistic stochastic control with geometric measure theory, offering a unified framework for understanding the asymptotic behavior of random paths.
The core methodology involves representing relative entropy variationally, analyzing the asymptotics of path measures driven by processes like Brownian motion, and establishing the Gamma-convergence of convex functionals on measure spaces. The results are validated through explicit models, such as Gaussian diffusion and jump processes, confirming the theoretical predictions with quantitative bounds. These findings have significant implications for fields ranging from machine learning to physics, where distribution matching and path optimization are central.
The study also introduces new analytical tools for measure-theoretic Gamma-convergence, filling gaps in the current literature. While the results are robust under the assumptions of large deviation principles, extending them to more complex or non-stationary systems remains an open challenge. Future directions include algorithm development for high-dimensional problems, broader class of stochastic dynamics, and applications in data science. Overall, this work advances the theoretical foundation of stochastic optimal transport, opening pathways for innovative computational and analytical approaches.
Deep Dive
Abstract
The aim of this article is to show that the Monge-Kantorovich problem is the limit of a sequence of entropy minimization problems when a fluctuation parameter tends down to zero. We prove the convergence of the entropic values to the optimal transport cost as the fluctuations decrease to zero, and we also show that the limit points of the entropic minimizers are optimal transport plans. We investigate the dynamic versions of these problems by considering random paths and describe the connections between the dynamic and static problems. The proofs are essentially based on convex and functional analysis. We also need specific properties of Gamma-convergence which we didn't find in the literature. Hence we prove these Gamma-convergence results which are interesting in their own right.