From plans to maps: Nonlocal regularization of optimal transport
Abstract
We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals Γ-converge to the classical Kantorovich problem, while the first-order Γ-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.
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BibTeX
@misc{carioni2026from,
title = {{From plans to maps: Nonlocal regularization of optimal transport}},
author = {Marcello Carioni and Leonardo Del Grande and José A. Iglesias and Hidde Schönberger},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05062},
eprint = {2609.05062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05062},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From plans to maps: Nonlocal regularization of optimal transport
AU - Marcello Carioni
AU - Leonardo Del Grande
AU - José A. Iglesias
AU - Hidde Schönberger
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05062
UR - https://arxiv.org/abs/2609.05062
AB - We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals Γ-converge to the classical Kantorovich problem, while the first-order Γ-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.
ER -