[Paper Review] A survey of the Schr\\"odinger problem and some of its connections with optimal transport

C. Léonard|arXiv (Cornell University)|Aug 1, 2013

Spectral Theory in Mathematical Physics49 references348 citations

TL;DR

This paper presents a comprehensive survey of the Schrödinger problem, a stochastic optimal transport problem that minimizes relative entropy subject to marginal constraints. It establishes a deep connection between the Schrödinger problem and optimal transport by showing that the dynamic Schrödinger problem is equivalent to a static problem involving Brownian bridges and that its solution links to the quadratic Monge-Kantorovich transport problem through the principle of least action.

ABSTRACT

This article is aimed at presenting the Schr\\"odinger problem and some of its connections with optimal transport. We hope that it can be used as a basic user's guide to Schr\\"odinger problem. We also give a survey of the related literature. In addition, some new results are proved.

Motivation & Objective

• To present the Schrödinger problem as a stochastic optimal transport problem minimizing relative entropy under marginal constraints.
• To clarify the relationship between the dynamic Schrödinger problem and the static Schrödinger problem via disintegration of measures.
• To establish connections between the Schrödinger problem and the classical Monge-Kantorovich optimal transport problem with quadratic cost.
• To provide a rigorous treatment of relative entropy with respect to unbounded measures, essential for defining the Schrödinger problem on path spaces.
• To serve as a user's guide and literature survey for researchers entering the field of stochastic optimal transport and large deviations.

Proposed method

• Formulates the dynamic Schrödinger problem as minimizing relative entropy $ H(P|R) $ over probability measures $ P $ on path space $ \Omega $, constrained by fixed initial and final marginals $ \mu_0, \mu_1 $.
• Uses the disintegration formula $ \widehat{P} = \int R^{xy} \widehat{\pi}(dxdy) $, where $ R^{xy} $ is the Brownian bridge measure and $ \widehat{\pi} $ solves the static Schrödinger problem on $ \mathcal{X} \times \mathcal{X} $.
• Shows that the static Schrödinger problem $ H(\pi|R_{01}) \to \min $ with $ \pi_0 = \mu_0, \pi_1 = \mu_1 $ is equivalent to the dynamic problem, with $ R_{01}(dxdy) \propto \exp(-d(x,y)^2/2) \, \textrm{vol}(dx)\textrm{vol}(dy) $.
• Establishes equivalence between the Schrödinger problem and the quadratic Monge-Kantorovich optimal transport problem via the dynamic cost $ C(\omega) = \int_0^1 |\dot{\omega}_t|^2/2 \, dt $.
• Introduces a rigorous definition of relative entropy with respect to unbounded measures using a weight function $ W $, ensuring $ H(p|r) $ is well-defined when $ \int W \, dp < \infty $.
• Applies the variational identity $ H(p|r) = \sup \left\{ \int u \, dp - \log \int e^u \, dr \right\} $ to characterize relative entropy in terms of cumulant generating functions.

Experimental results

Research questions

• RQ1 How does the Schrödinger problem relate to optimal transport, and what is the precise mathematical connection between the two?
• RQ2 What is the role of the Brownian bridge in the disintegration of the solution to the dynamic Schrödinger problem?
• RQ3 How can relative entropy be rigorously defined with respect to an unbounded reference measure, such as the law of Brownian motion on a non-compact manifold?
• RQ4 In what sense is the Schrödinger problem a stochastic regularization of the Monge-Kantorovich problem?
• RQ5 What are the implications of the equivalence between the dynamic Schrödinger problem and the static problem for large deviations and statistical physics?

Key findings

• The solution $ \widehat{P} $ to the dynamic Schrödinger problem disintegrates as a mixture of Brownian bridges $ R^{xy} $, with the mixing measure $ \widehat{\pi} $ solving the static Schrödinger problem on $ \mathcal{X} \times \mathcal{X} $.
• The values of the dynamic and static Schrödinger problems are equal: $ \inf \eqref{sdyn} = \inf \eqref{s} $, establishing their equivalence.
• The static Schrödinger problem is equivalent to the quadratic Monge-Kantorovich optimal transport problem with cost $ c(x,y) = d(x,y)^2/2 $, linking it to classical optimal transport.
• The dynamic cost $ C(\omega) = \int_0^1 |\dot{\omega}_t|^2/2 \, dt $ achieves its minimum on paths from $ x $ to $ y $ along constant-speed geodesics, and the solution to (MK dyn) corresponds to deterministic paths $ \gamma^{xy} $.
• The relative entropy $ H(p|r) $ is well-defined for $ p \in \mathrm{P}(Y) $ with $ \int W \, dp < \infty $, using a weight function $ W $ satisfying $ \int e^{-W} \, dr < \infty $, ensuring coherence across different choices of $ W $.
• The variational formula $ H(p|r) = \sup \left\{ \int u \, dp - \log \int e^u \, dr \right\} $ holds for measurable functions $ u $ with $ \sup |u|/W < \infty $, and characterizes $ H(\cdot|r) $ as a convex, lower semicontinuous function.