Théo Dumont

2026

1. preprint

   

   Learning Monge maps by lifting and constraining Wasserstein gradient flows

   Théo Dumont, Théo Lacombe, and François-Xavier Vialard

   preprint, 2026

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   We study the estimation of optimal transport (OT) maps between an arbitrary source probability measure and a log-concave target probability measure. Our contributions are twofold. First, we propose a new evolution equation in the set of transport maps. It can be seen as the gradient flow of a lift of some user-chosen divergence (e.g., the KL divergence, or relative entropy) to the space of transport maps, constrained to the convex set of optimal transport maps. We prove the existence of long-time solutions to this flow as well as its convergence toward the OT map as time goes to infinity, under standard convexity conditions on the divergence. Second, we study the practical implementation of this constrained gradient flow. We propose two time-discrete computational schemes-one explicit, one implicit-, and we prove the convergence of the latter to the OT map as time goes to infinity. We then parameterize the OT maps with convexity-constrained neural networks and train them with these discretizations of the constrained gradient flow. We show that this is equivalent to performing a natural gradient descent of the lift of the chosen divergence in the neural networks’ parameter space. Empirically, our scheme outperforms the standard Euclidean gradient descent methods used to train convexity-constrained neural networks in terms of approximation results for the OT map and convergence stability, and it still yields better results than the same approach combined with the widely used adam optimizer.

   @article{dumont2026learning,
     title = {Learning Monge maps by lifting and constraining Wasserstein gradient flows},
     author = {Dumont, Th{\'e}o and Lacombe, Th{\'e}o and Vialard, Fran{\c{c}}ois-Xavier},
     year = {2026},
     journal = {preprint},
   }

2024

1. FoCMBest poster

   

   On the existence of Monge maps for the Gromov-Wasserstein problem

   Théo Dumont, Théo Lacombe, and François-Xavier Vialard

   Foundations of Computational Mathematics, 2024

   Best poster award at Geometric Sciences in Action, 2024.

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   The Gromov–Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between points. Akin to the standard optimal transportation problem, it is natural to ask for conditions guaranteeing some structure on the optimizers, for instance if these are induced by a (Monge) map. We study this question in Euclidean spaces when the cost functions are either given by (i) inner products or (ii) squared distances, two standard choices in the literature. We establish the existence of an optimal map in case (i) and of an optimal 2-map (the union of the graphs of two maps) in case (ii), both under an absolute continuity condition on the source measure. Additionally, in case (ii) and in dimension one, we numerically design situations where optimizers of the Gromov–Wasserstein problem are 2-maps but are not maps. This suggests that our result cannot be improved in general for this cost. Still in dimension one, we additionally establish the optimality of monotone maps under some conditions on the measures, thereby giving insight on why such maps often appear to be optimal in numerical experiments.

   @article{dumont2022existence,
     title = {On the existence of Monge maps for the Gromov-Wasserstein problem},
     author = {Dumont, Th{\'e}o and Lacombe, Th{\'e}o and Vialard, Fran{\c{c}}ois-Xavier},
     year = {2024},
     journal = {Foundations of Computational Mathematics},
   }