Théo Dumont

I am a PhD student in optimal transport at the LIGM (Université Gustave Eiffel) under the supervision of François-Xavier Vialard, Théo Lacombe and Virginie Ehrlacher. Just before that, I worked with Philipp Harms (NTU Singapore) and Klas Modin (Chalmers University) on some topics related to the infinite-dimensional geometry of optimal transport.

My interests span the theory of optimal transport (OT), infinite-dimensional Riemannian geometry and machine learning, and I am passionate about the interplay between those fields. I am currently working on the geometry of several OT-related problems, such as Gromov–Wasserstein, on gradient flows on the space of probability measures, and on the geometry of neural networks. I am always happy to discuss, so feel free to contact me!

Contact: theo.dumont [at] univ-eiffel [dot] fr
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News

Oct 29, 2024	I gave a talk at the Congrès des Jeunes Chercheur.es en Mathématiques Appliquées (CJC-MA) in Lyon, on the existence of Monge maps for the Gromov–Wasserstein problem. [slides, paper]
Oct, 2024	I wrote some notes on logarithmic Sobolev inequalities and related topics for a short tutorial in the New Monge problems working group at LIGM. They are available here.
Aug 7, 2024	I presented a poster at the Machine Learning in Infinite Dimensions workshop in Bath, on the existence of Monge maps for the Gromov–Wasserstein problem. [poster, paper]
May 29, 2024	I presented a poster at the Geometric Sciences in Action conference in Marseille, on the existence of Monge maps for the Gromov–Wasserstein problem. It won the competition! [poster, paper]
Jan 19, 2024	I gave a talk at the MAP5 lab PhDs’ seminar about the existence of Monge maps for the Gromov–Wasserstein problem. [slides, paper]

Selected publication(s)

2024

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.

Bib Abs PDF Slides Poster Code

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}, }