- 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.

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.

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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An acceleration of the Sinkhorn algorithm for unbalanced transport is proposed and Frank-Wolfe to solve the 1D unbalanced OT problem.

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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.

In this talk, I present two very different applications related by the simple idea of invertible transformations.
- The first topic is an analysis of inverse consistency penalty in image matching in conjunction with the use of neural networks. We show that neural networks favours the emergence of smooth transformation for the inverse consistency. Experimentally, we show that this behaviour is fairly stable with respect to the chosen architecture. This is joint work with H. Greer, R. Kwitt and M. Niethammer.
-The second topic is an analysis of global convergence of residual networks when the residual block is parametrized via reproducing kernel Hilbert space vector field. We prove that the resulting problem satisfies the so-called Polyak-Lojasiewicz property, for instance ensuring global convergence if the iterates are bounded. We show that this property applies in a continuous limit as well as in the fully discrete setting. This is joint work with R. Barboni and G. Peyré.

I explain the formulation of the Wasserstein-Fisher-Rao distance, introduced a few years ago as the natural extension of the Wasserstein L2 metric to the space of positive Radon measures. We present the equivalence between the dynamic formulation and a static formulation, where we relax the marginal constraints with relative entropy. Then, we will present a connection with standard optimal transport on a cone space. The second part of the talk is motivated by this optimal transport distance and we study a generalized Camassa-Holm equation, for which we study existence of minimizing generalized geodesics, a la Brenier.

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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