Séance du 5 février 2024

Séance organisée par Umut Şimşekli  et Sylvain Le Corff 

Lieu : IHP,  salle 314 (Grisvard)

14.00 : Gabriel Victorino Cardoso (CMAP) 

Titre :  Monte Carlo guided Diffusion for Bayesian linear inverse problems

Résumé : Ill-posed linear inverse problems arise frequently in various applications, from computational photography to medical imaging. A recent line of research exploits Bayesian inference with informative priors to handle the ill-posedness of such problems. Amongst such priors, score-based generative models (SGM) have recently been successfully applied to several different inverse problems. In this talk, we exploit the particular structure of the prior defined by the SGM to define a sequence of intermediate linear inverse problems. As the noise level decreases, the posteriors of these inverse problems get closer to the target posterior of the original inverse problem. To sample from this sequence of posteriors, we propose the use of Sequential Monte Carlo (SMC) methods. The proposed algorithm, MCGDiff, is shown to be theoretically grounded and we provide numerical simulations showing that it outperforms competing baselines when dealing with ill-posed inverse problems in a Bayesian setting.

15.00 : James Thornton (University of Oxford)

Titre : Flow / Diffusion Based Generative Modelling for and with Optimal Transport

Résumé : Diffusion models; Schrodinger bridges and flow matching are interconnected paradigms which provide a promising new approach to high dimensional optimal transport. These methods leverage the scalability of modern generative models but may utilise data-to-data couplings for applications in various fields including inverse problems; text to speech; molecular dynamics, and single cell biology. The usefulness of these couplings depends on the choice of reference dynamics, and hence corresponding ground cost in the transport problem. There are however many common misunderstandings and open challenges remaining; in particular around applicability to general reference dynamics. This talk will introduce and contrast the relative merits of diffusion Schrodinger bridges and flow matching methods, discuss choices in reference dynamics including an extension to the Riemannian manifold setting; and related computational challenges.

16.00 : Antonio Ocello (Ecole Polytechnique)

Titre : Optimal tuning of the noise schedule for Diffusion-based Models

Résumé : Score-based generative models (SGMs) aim at estimating a target data distribution by learning score functions using only noise-perturbed samples from the target. Recent literature has focused extensively on assessing the error between the target and estimated distributions, gauging the generative quality through the Kullback-Leibler (KL) divergence and Wasserstein distances.  All existing results  have been obtained so far for time-homogeneous speed of the noise schedule. Under mild assumptions on the data distribution, we establish an upper bound for the KL divergence between the target and the estimated distributions, explicitly depending on any time-dependent noise schedule. Assuming that the score is Lipschitz continuous, we provide an improved error bound in Wasserstein distance, taking advantage of favourable underlying contraction mechanisms. We also propose an algorithm to automatically tune the noise schedule using the proposed upper bound. We illustrate empirically the performance of the noise schedule optimization in comparison to standard choices in the literature.