[Paper Review] The Monge Problem for distance cost in geodesic spaces

Stefano Bianchini, Fabio Cavalletti|arXiv (Cornell University)|Mar 14, 2011

Geometry and complex manifolds15 references4 citations

TL;DR

This paper establishes conditions under which the Monge problem admits a solution in geodesic metric spaces with distance cost, by reducing the transport problem to one-dimensional problems along geodesics. The key contribution is proving that if the disintegration of the source measure along geodesics is absolutely continuous with respect to 1D Hausdorff measure, then a transport map exists, even when $d_L$-cyclical monotonicity alone is insufficient for optimality.

ABSTRACT

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X,dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the transport problem π which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w.r.t. the 1- dimensional Hausdorff distance induced by dL. It is known that this regularity is sufficient for the construction of a transport map. We study also the dynamics of transport along the geodesic, the stability of our conditions and show that in this setting dL-cyclical monotonicity is not sufficient for optimality.

Motivation & Objective

• To resolve the Monge problem in geodesic metric spaces where the cost is given by a geodesic distance $d_L$.
• To identify conditions under which $d_L$-cyclical monotonicity implies optimality, despite its insufficiency in general.
• To reduce the multidimensional optimal transport problem to one-dimensional transport problems along geodesics.
• To establish the existence of a transport map by ensuring the disintegration of the source measure along geodesics is absolutely continuous with respect to 1D Hausdorff measure.

Proposed method

• The authors partition the transport set $\mathcal{T}_e$ into geodesic rays using the non-branching assumption and cyclical monotonicity of the transport plan $\pi$, forming a partition $R$ of the set $\mathcal{T}$ of inner points of geodesics.
• They define a ray map $g = g^+ \cup g^-$ that assigns to each point in $\mathcal{T}$ the unique geodesic segment through it, enabling the decomposition of the transport problem into one-dimensional components.
• The disintegration of the source measure $\mu$ along each geodesic is analyzed, and two key assumptions are introduced to ensure the conditional probabilities are absolutely continuous with respect to the 1D Hausdorff measure.
• A dynamic interpretation is developed via the current $\dot{g}$, representing the flow along geodesics, and the transport equation is formulated to describe the evolution of sets under this flow.
• Stability of the non-degeneracy condition is analyzed through approximations of metric spaces, ensuring robustness of the results under perturbations.
• The paper constructs a transport map by combining selection principles with the regularity of disintegration, proving existence under the stated conditions.

Experimental results

Research questions

• RQ1 Under what conditions on the geodesic space and transport plan does the Monge problem admit a solution when the cost is given by a geodesic distance?
• RQ2 Can the multidimensional optimal transport problem be reduced to one-dimensional transport problems along geodesics in non-branching geodesic spaces?
• RQ3 Is $d_L$-cyclical monotonicity sufficient for optimality in geodesic spaces, or are additional regularity conditions required?
• RQ4 What conditions ensure that the disintegration of the source measure along geodesics is absolutely continuous with respect to the 1D Hausdorff measure?
• RQ5 How does the dynamic evolution of sets along geodesics, described by the current $\dot{g}$, relate to the stability and existence of transport maps?

Key findings

• The Monge problem has a solution in non-branching geodesic spaces when the disintegration of the source measure $\mu$ along geodesics is absolutely continuous with respect to the 1D Hausdorff measure induced by $d_L$.
• The reduction of the transport problem to one-dimensional problems along geodesics is valid under the assumption that geodesics are $d$-continuous and locally compact.
• The $d_L$-cyclical monotonicity of a transport plan is not sufficient for optimality, as demonstrated by a counterexample in a metric space with a specific structure.
• The disintegration of $\mu$ is strongly consistent if and only if the conditional probabilities are absolutely continuous with respect to the 1D Hausdorff measure on each geodesic.
• The stability of the non-degeneracy condition is preserved under approximation by metric spaces, ensuring robustness of the solution under perturbations.
• A transport map exists when the disintegration of $\mu$ along geodesics is continuous or absolutely continuous, as guaranteed by the two introduced assumptions on the transport plan $\pi$.