Nous proposons une introduction au cadre du transport optimal, qui permet de définir une métrique sur l’espace des mesures de probabilité sur un espace métrique mesurable.
Cette métrique permet de donner un sens à la notion de flot de gradient, malgré l’absence de structure hilbertienne, et fournit un cadre théorique à un grand nombre d’équations d’évolution classiques, cadre qui privilégie...
Diffusion Flow Matching (DFM) models provide a powerful framework for generative modeling, and recent research suggests their connection to the entropic optimal transport (EOT) problem. In this talk, I will focus on the theoretical foundations of DFMs, their connections with EOT, and present novel non-asymptotic guarantees for these models. Specifically, I will discuss how, under mild...
Functional inequalities (Poincaré, log-Sobolev, etc.) represent a ubiquitous tool in probability, since they help us to quantify the convergence to equilibrium of ergodic Markov processes and imply good concentration properties of a probability measure, among other properties. It is natural to wonder if these inequalities remain valid if we perturb the measure. It is known that if there exists...
In 1998, Jordan, Kinderlehrer and Otto proposed an approximation scheme for the Fokker-Planck equation that gives a very elegant interpretation of this PDE as a gradient flow in a certain metric space : the Wasserstein space. Thanks to this publication, the PDE community discovered a way of using optimal transport tools and gradient flow theory for a large class of PDEs, raising a certain...
