Kantorovich Distance via Spanning Trees: Properties and Algorithms

TL;DR

This work studies optimal transport between probability measures on a finite metric space with ground cost $d_G$ induced by a connected graph $G$, using a reformulation that minimizes over rooted spanning trees $T$ of $G$. It derives an explicit Kantorovich potential on the optimal tree $T_*$ under a weak non-degeneracy assumption, and provides a dynamic-programming algorithm to construct an optimal transport plan on $T_*$ along with a corresponding potential; it also introduces a simulated annealing approach to identify the optimal spanning tree itself. The theoretical contributions connect the Kantorovich distance to the Arens--Eells norm, establish duality and plan structure, and show how the tree case yields closed-form expressions for both the AE norm and the OT distance. Numerical experiments on grid graphs derived from image data illustrate the methods, demonstrate convergence of the SA procedure, and highlight the geometric interpretation that supports mass transport along geodesics of the original graph. The results offer a computational framework for OT on finite metric spaces via spanning-tree decompositions, with potential applications to large-scale graph-based transport problems and related dual formulations.

Abstract

We study optimal transport between probability measures supported on the same finite metric space, where the ground cost is a distance induced by a weighted connected graph. Building on recent work showing that the resulting Kantorovich distance can be expressed as a minimization problem over the set of spanning trees of this underlying graph, we investigate the implications of this reformulation on the construction of an optimal transport plan and a dual potential based on the solution of such an optimization problem. In this setting, we derive an explicit formula for the Kantorovich potential in terms of the imbalanced cumulative mass (a generalization of the cumulative distribution in R) along an optimal spanning tree solving such a minimization problem, under a weak non-degeneracy condition on the pair of measures that guarantees the uniqueness of a dual potential. Our second contribution establishes the existence of an optimal transport plan that can be computed efficiently by a dynamic programming procedure once an optimal spanning tree is known. Finally, we propose a stochastic algorithm based on simulated annealing on the space of spanning trees to compute such an optimal spanning tree. Numerical experiments illustrate the theoretical results and demonstrate the practical relevance of the proposed approach for optimal transport on finite metric spaces.

Figures (9)

• Figure 1: The red dots represent a set of $N=6$ vertices, and the blue segments are the edges in $T_\ast$. The green and dashed arrow is an oriented edge $(x,y)$ in the support of $\gamma_{T_\ast}$ (that is such that $\gamma_{T_\ast}(x,y)>0$) which does not belong to $\mathcal{E}_{T_\ast}$. The mass sent from $x$ to $y$ follows the blue path.
• Figure 2: Values of the weights $\mu$ and $\nu$ with $N=6$. The graph $G$ is a line tree with 6 spanning trees which have the same structure with only the root placed at different vertices.
• Figure 3: Illustration of Equations \ref{['eq:pos']} and \ref{['eq:neg']} from Proposition \ref{['prop:optim_plan']}, which says that the transit of mass can be made only in one direction. The red path represents a positive transport $\gamma_T(x,y) > 0$ and the dashed path represent a transport that is not allowed since $\gamma_T(u,v)=0$.
• Figure 4: Probability measures $\mu$ and $\nu$ supported on a regular two-dimensional lattice $G$ of size $p \times p$, with $p = 7$, in both the noiseless and noisy settings. In all the figures, the red dots represent the set $\mathcal{X}$ of vertices. The edges of the graph $G$ are shown as vertical and horizontal blue segments.
• Figure 5: Probability measures $\mu$ and $\nu$ supported on a regular two-dimensional lattice $G$ of size $p \times p$, with $p = 14$, in both the noiseless and noisy settings. In all the figures, the red dots represent the set $\mathcal{X}$ of vertices. The edges of the graph $G$ are shown as vertical and horizontal blue segments.

Theorems & Definitions (49)

• Definition 1.1
• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Definition 2.1
• Proposition 2.1: Theorem 4 in KW68
• Remark 2.1
• Proposition 2.2
• proof
• Remark 2.2