[Paper Review] FROM THE SCHRODINGER PROBLEM TO THE MONGE-KANTOROVICH PROBLEM

Christian Léonard|arXiv (Cornell University)|Aug 16, 2016

Geometry and complex manifoldsMathematics29 references207 citations

TL;DR

This paper establishes a rigorous connection between the Schrödinger problem—entropy minimization under stochastic dynamics—and the classical Monge-Kantorovich optimal transport problem. By showing that the entropic minimizers converge to optimal transport plans as the fluctuation parameter tends to zero, it proves that the Monge-Kantorovich cost emerges as the large-deviation limit of Schrödinger's entropic problem, using Γ-convergence and large deviation principles in a functional analytic framework.

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.

Motivation & Objective

• . To establish the convergence of entropic minimizers to optimal transport plans as the fluctuation parameter vanishes.
• . To show that the optimal transport cost arises as the large-deviation limit of entropic costs.
• . To unify dynamic (path-based) and static (marginal) formulations of optimal transport through probabilistic and variational methods.
• . To develop and prove novel Γ-convergence results for convex functions and constrained minimization problems in weakly compact spaces.
• . To provide a functional-analytic framework that makes probabilistic concepts accessible to analysts without requiring deep knowledge of stochastic processes or large deviation theory.

Proposed method

• . Uses Γ-convergence to analyze the limit of entropy minimization problems as the fluctuation parameter k → ∞.
• . Employs large deviation principles (LDP) to characterize the asymptotic behavior of stochastic processes Rk with decreasing variance.
• . Establishes a correspondence between the rate function C(ω) of the LDP and the transport cost c(x,y) = inf{C(ω) : ω0=x, ω1=y}.
• . Proves that the minimizers of the entropic problem (2) converge weakly to optimal transport plans for the limit cost c.
• . Applies the contraction principle and Laplace-Varadhan principle to derive the LDP for empirical measures of initial and final positions.
• . Uses convex and functional analysis to prove equi-coercivity and Γ-liminf/limsup conditions for convergence of functionals.

Experimental results

Research questions

• RQ1 . How does the solution to Schrödinger's entropic problem converge to the solution of the Monge-Kantorovich optimal transport problem as fluctuations diminish?
• RQ2 . What is the precise relationship between the rate function of a stochastic process and the resulting transport cost in the limit?
• RQ3 . Can the dynamic formulation of optimal transport (via random paths) be rigorously linked to the static formulation (via joint measures) using large deviations?
• RQ4 . Under what conditions does the sequence of entropic minimizers converge to an optimal transport plan?
• RQ5 . What functional-analytic tools—specifically Γ-convergence—are required to prove the convergence of these variational problems?

Key findings

• . The entropic cost values converge to the Monge-Kantorovich optimal transport cost as the fluctuation parameter k → ∞.
• . The limit points of the entropic minimizers are optimal transport plans for the limiting cost c(x,y) = inf{C(ω) : ω0=x, ω1=y}.
• . For Brownian motion with diffusion coefficient 1/k, the limiting cost is c(x,y) = 1/2|y−x|², and the minimizers converge to the displacement interpolation between µ0 and µ1.
• . When the quadratic cost problem has a unique solution, the sequence of entropic minimizers Pk converges weakly to the deterministic process bP = ∫ δσxy bπ(dxdy), where bπ is the optimal transport plan.
• . The paper proves new Γ-convergence results for constrained minimization problems under equi-coercivity and continuity assumptions, which are essential for the main convergence theorems.
• . The contraction principle and Laplace-Varadhan principle are used to derive the LDP for the joint law of initial and final positions, yielding the transport cost as the rate function.