- Optimal transport; applications and numerics.
- Diffeomorphic flows; machine learning and medical image registration applications.
- Calculus of variations and geometry; applications to shape spaces and fluid flows.
- Applied and computational mathematics.

When trying to understand the low temperature asymptotic of entropic regularization in optimal transport, Gibbs measures naturally …

In a couple of recent papers on applied optimal transport (OT), we used the fact that the so-called semi-dual formulation of OT …

In a project with a student, I recently derived a non-local diffusion PDE which turns out, surprisingly for me, to have a name: Stein …

Complete list of publications here.

We give a geometric formulation of the first-order term of the Laplace method in a particular case. Our main result expresses the …

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An easy to read review of extension of optimal transport to positive measures, with an emphasis on numerics and entropic …

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In Euclidean space, we prove existence of optimal Monge maps for the case of the inner product cost and for the case of the quadratic …

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Learning with gradient inverse consistency and no explicit regularization achieves better diffeomorphic registration performances than …

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We derive stability estimates on the semi-dual formulation of unbalanced optimal transport and propose an optimization algorithm, …

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Toric geometry of entropic regularization is explored with a focus on entropic conic unbalanced optimal transport. Generalized …

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This talk has two parts. First we present a possible extension of the Gromov-Wasserstein problem to the setting of metric measures spaces, whose total mass is not necessarily equal to 1. We propose a true distance and a lower bound which is more friendly for computations. Second, we study the existence of Monge maps as optimizer of the standard Gromov-Wasserstein problem for two different costs in euclidean spaces. The first cost for which we show existence of Monge maps is the scalar product, the second cost is the quadratic cost between the squared distances for which we show the structure of a bi-map. We present numerical evidence that the last result is sharp.

This talk has two parts. First we present a possible extension of the Gromov-Wasserstein problem to the setting of metric measures spaces, whose total mass is not necessarily equal to 1. We propose a true distance and a lower bound which is more friendly for computations. Second, we study the existence of Monge maps as optimizer of the standard Gromov-Wasserstein problem for two different costs in euclidean spaces. The first cost for which we show existence of Monge maps is the scalar product, the second cost is the quadratic cost between the squared distances for which we show the structure of a bi-map. We present numerical evidence that the last result is sharp.

We show how to break the curse of dimension for the estimation of optimal transport distance between two smooth distributions for the Euclidean squared distance. The approach relies on essentially one tool: represent inequality constraints in the dual formulation of OT by equality constraints with a sum of squares in reproducing kernel Hilbert space. By showing this representation is tight in the variational formulation, one can then leverage smoothness to break the curse. (*) However, the constants associated with the algorithm a priori scale exponentially with the dimension.

Member of the research lab LIGM (Laboratoire d’informatique Gaspard Monge) and teaching signal processing.

Member of the research lab Ceremade, UMR CNRS 7534, and teaching applied mathematics.

Member of the Institute for Mathematical Sciences and the Math department.

Optimal transport, Diffeomorphisms and applications to imaging

Hamiltonian formulation of diffeomorphic image matching

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