Entropic regularization of Monge's problem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908996763385856 |
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| author | Nutz, Marcel Zhong, Chenyang |
| author_facet | Nutz, Marcel Zhong, Chenyang |
| contents | We study the vanishing-regularization limit of entropically regularized optimal transport (EOT) for the Euclidean distance cost $c(x,y)=\|x-y\|$ in dimension $d>1$. We develop a comprehensive variational convergence framework that entails two main results. First, we resolve the longstanding entropic selection problem: the EOT minimizer converges to a distinguished optimal transport plan that is characterized explicitly as the solution of a constrained EOT problem on each transport ray. Denoting by $\varepsilon>0$ the regularization parameter, this selection holds for all $o(\varepsilon)$-approximate minimizers, with sharp failure at the $O(\varepsilon)$ scale. Second, we establish an explicit second-order expansion of the entropic transport cost. The second-order term encodes the geometry of the regularization and reveals the optimal asymptotic tradeoff between entropy and transport cost. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Entropic regularization of Monge's problem Nutz, Marcel Zhong, Chenyang Optimization and Control Analysis of PDEs Functional Analysis Probability We study the vanishing-regularization limit of entropically regularized optimal transport (EOT) for the Euclidean distance cost $c(x,y)=\|x-y\|$ in dimension $d>1$. We develop a comprehensive variational convergence framework that entails two main results. First, we resolve the longstanding entropic selection problem: the EOT minimizer converges to a distinguished optimal transport plan that is characterized explicitly as the solution of a constrained EOT problem on each transport ray. Denoting by $\varepsilon>0$ the regularization parameter, this selection holds for all $o(\varepsilon)$-approximate minimizers, with sharp failure at the $O(\varepsilon)$ scale. Second, we establish an explicit second-order expansion of the entropic transport cost. The second-order term encodes the geometry of the regularization and reveals the optimal asymptotic tradeoff between entropy and transport cost. |
| title | Entropic regularization of Monge's problem |
| topic | Optimization and Control Analysis of PDEs Functional Analysis Probability |
| url | https://arxiv.org/abs/2604.21578 |