Théo Dumont

I am a research intern in infinite-dimensional Riemannian geometry applied to deep learning at NTU Singapore, working with Pr. Philipp Harms. I will start my PhD at the LIGM (Université Gustave Eiffel) in late 2023. I recently completed a dual-degree MSc at Mines Paris - PSL and ENS Paris-Saclay (MVA).

I am passionate about mathematics, in particular about geometry, optimal transport and deep learning, and I am always very happy to discuss. Please feel free to contact me!

News

Nov, 2022	I joined NTU Singapore for a research experience in infinite-dimensional Riemannian geometry applied to deep learning under the supervision of Philipp Harms.
Oct, 2022	Our preprint On the existence of Monge maps for the Gromov-Wasserstein problem with Théo Lacombe and François-Xavier Vialard is available on arxiv.
Apr, 2022	I joined the LIGM at Université Gustave Eiffel for a research internship in optimal transport under the supervision of François-Xavier Vialard and Théo Lacombe.

Selected publications

On the Existence of Monge Maps for the Gromov-Wasserstein Problem

Théo Dumont, Théo Lacombe, and François-Xavier Vialard

Under review, 2022

Bib Abs PDF Code

In this work, we study the structure of minimizers of the quadratic Gromov–Wasserstein (GW) problem on Euclidean spaces for two different costs. The first one is the scalar product for which we prove that it is always possible to find optimizers as Monge maps and we detail the structure of such optimal maps. The second cost is the squared Euclidean distance for which we show that the worst case scenario is the existence of a bi-map structure. Both results are direct and indirect consequences of an existence result of optimal maps in the standard optimal transportation problem for costs that are defined by submersions. In dimension one for the squared Euclidean distance, we show numerical evidence for a negative answer to the existence of a Monge map under the conditions of Brenier’s theorem, suggesting that our result cannot be improved in general. In addition, we show that a monotone map is optimal in some non-symmetric situations, thereby giving insight on why such a map often appears 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 = {2022}, journal = {Under review}, }