Discrete Approximation of Optimal Transport on Compact Spaces

TL;DR

This work develops a general, fully discrete framework for approximating the Kantorovich and Monge formulations of optimal transport on general compact spaces using $h$-partitions to build discrete marginals and a finite cost LP. A novel element is the extension of projection maps to arbitrary metric spaces via geometric medians (or barycenters) and semidiscrete constructions, enabling map approximation without linear structure. The authors prove sharp convergence results: discrete plans converge to the Kantorovich minimizers, optimal values converge with an explicit modulus $\omega_c(h)$, and projection maps converge to the true OT map under mild (strong) uniqueness conditions; they also quantify when fully discrete convergence in the disc_p metric holds, relating it to the μ-nullity of the map’s discontinuity set. The results substantially broaden the theory of OT discretization beyond Euclidean settings, with implications for robust numerical schemes and convergence guarantees in general compact spaces.

Abstract

We investigate the approximation of Monge--Kantorovich problems on general compact metric spaces, showing that optimal values, plans and maps can be effectively approximated via a fully discrete method. First we approximate optimal values and plans by solving finite dimensional discretizations of the corresponding Kantorovich problem. Then we approximate optimal maps by means of the usual barycentric projection or by an analogous procedure available in general spaces without a linear structure. We prove the convergence of all these approximants in full generality and show that our convergence results are sharp.

Figures (5)

• Figure 1: The thick lines show the zero set of the cost $c$ and the dots inside the gray squares show $\operatorname{supp}({\pi_k})$. In \ref{['figure_example_no_convergence_semicontinuous_cost']}, note that all points are off the diagonal. In \ref{['figure_example_only_weak_convergence_for_pi_k']}, note that the points off the diagonal "do the work" of the two missing points in $E_1\times F_1$ and $E_4\times F_4$.
• Figure 2: The supports of $\pi_h$ (left), ${\pi_h^{\mspace{2mu}s}}$ (right) and ${\pi_h^{\mspace{2mu}c}}$ (shaded region).
• Figure 3: The measure ${\pi_h^{\mspace{2mu}s}}$ in \ref{['figure_semidiscrete_pi_h_supports']} disintegrates with respect to $\mu$ as a family of finitely supported measures $(\gamma_x)_{x\in X}$. For $A$ and $x_0$ as above we get $(\delta_{x_0} \otimes \gamma_{x_0} )[A] = \gamma_{x_0}[A_{x_0}] = \tfrac{\pi_{21}}{\mu_2} + \tfrac{\pi_{23}}{\mu_2}$.
• Figure 4: In \ref{['sub@figure_example_non_convergence_multiple_maps']} the thick lines show the zero set $Z_c$ of the cost $c$ and the dots inside the gray squares show $\operatorname{supp}({\pi_k})$. In \ref{['sub@figure_example_no_convergence_maps_KU_and_MU_but_not_SU']} the supports of $\mu$, $\nu$ and $\pi_{T*}$ are plotted in black and the support of $\pi_*$ is plotted in dark gray.
• Figure 5: In \ref{['sub@figure_example_non_convergence_disc_p_plans']} the dots show $\operatorname{supp}({\pi_k})$. In \ref{['sub@figure_example_non_convergence_disc_p_maps']} the graph of $T_*$ is plotted in black and the graph of $T_k$ is plotted in dark gray.

Theorems & Definitions (66)

• Theorem A
• Theorem B
• Corollary B
• Remark 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Remark 2.6