On the relation between optimal transport and Schrödinger bridges: A stochastic control viewpoint (1412.4430v1)

Abstract: We take a new look at the relation between the optimal transport problem and the Schr\"{o}dinger bridge problem from the stochastic control perspective. We show that the connections are richer and deeper than described in existing literature. In particular: a) We give an elementary derivation of the Benamou-Brenier fluid dynamics version of the optimal transport problem; b) We provide a new fluid dynamics version of the Schr\"{o}dinger bridge problem; c) We observe that the latter provides an important connection with optimal transport without zero noise limits; d) We propose and solve a fluid dynamic version of optimal transport with prior; e) We can then view optimal transport with prior as the zero noise limit of Schr\"{o}dinger bridges when the prior is any Markovian evolution. In particular, we work out the Gaussian case. A numerical example of the latter convergence involving Brownian particles is also provided.

• The paper presents an elementary derivation of the Benamou-Brenier formulation of optimal transport using a stochastic control approach.
• The paper introduces a novel fluid dynamic formulation for Schrödinger bridges, linking them to optimal transport without requiring vanishing diffusion.
• The paper formulates optimal transport with a prior evolution, demonstrating convergence of Schrödinger bridge solutions to OT solutions under a zero noise Gaussian scenario.

Overview of the Relation Between Optimal Transport and Schrödinger Bridges: A Stochastic Control Viewpoint

This paper provides a detailed investigation into the interconnections between optimal transport (OT) and Schrödinger bridge (SB) problems through a stochastic control lens. Previous studies have recognized the relationship between these two problems, but this paper brings new insights and extends the understanding of their connections. The authors present elementary derivations and new formulations that illuminate the depth and richness of these connections beyond existing literature.

Summary

Key contributions of the paper include:

1. Elementary Derivation of Optimal Transport: The authors present a straightforward derivation of the Benamou-Brenier (BB) fluid dynamic formulation of the classical Monge-Kantorovich OT problem. This derivation uses a stochastic control framework to interpret the transport problem as minimizing the energy associated with a flow, subject to continuity constraints.
2. New Perspective on Schrödinger Bridges: The paper introduces a novel fluid dynamic formulation of the SB problem. Unlike traditional approaches, this formulation recognizes important links between OT and SB without requiring the diffusion coefficient to vanish, broadening the applicability of these concepts.
3. Optimal Transport with Prior: A significant innovation is the formulation and resolution of a fluid dynamic version of OT that incorporates a prior evolution, as described by any Markovian processes. The authors show that this conceptually aligns OT with SB under a zero noise limit, particularly in the Gaussian scenario.

Numerical Results and Implications

• Numerical Example: An application involving the interpolation of overdamped Brownian particles' distributions demonstrates the convergence of SB solutions to OT solutions under diminishing noise. This highlights the practical utility of the theoretical developments presented in the paper.
• Theoretical Implications: The research extends foundational theories in both SB and OT. It suggests a unified framework, allowing statisticians and probabilistic theorists to tackle complex distribution evolution problems across various fields, including quantum mechanics and economics.

Practical and Theoretical Implications

• Stochastic Control and Information Theory: The paper underscores the intersection of OT and SB with stochastic control and information theoretic principles. These insights facilitate new methodologies for performing tasks such as distribution alignment and inference in noisy environments.
• Future Developments in AI: The methodologies described could significantly impact areas like generative modeling, offering new strategies for creating probabilistically grounded models capable of bridging diverse datasets and objectives.

In conclusion, this paper advances the understanding of OT and SB connections, laying the groundwork for future research and application in stochastic processes, control theory, and beyond. The inclusion of numerical examples and solid theoretical foundations presents a compelling case for the practical applicability and the further development of these insights.