Séance du 11 janvier 2010

Lundi 11 janvier 2010

Organisateurs: Jean-Michel Poggi et Jérôme Dedecker

14h00 Fréderic Chazal (INRIA Saclay)

Geometric inference for probability measures: extracting robust geometric information from noisy data.

Abstract: Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (Betti numbers, curvatures,...) of this subset from the approximating point cloud data. In recent years, it appeared that the study of distance functions allows to address many of these questions successfully. However, one of the main limitations of this framework is that it does not cope well with outliers nor with background noise. In this talk, we will show how to extend the framework of distance functions to overcome this problem. Replacing compact subsets by measures, we will introduce a notion of distance function to a probability distribution in $\R^n$. These functions share many properties with classical distance functions, which makes them suitable for inference purposes. In particular, by considering appropriate level sets of these distance functions, it is possible to associate in a robust way topological and geometric features to a probability measure. If time permits, we will also mention a few other potential applications of this framework.

15h00 Simon Foucart (Universite Pierre et Marie Curie, Laboratoire Jacques-Louis Lions)

Randomness in Compressive Sensing

Abstract: This talk will provide an overview of the emerging field of Compressive Sensing, which aims at recovering sparse signals from a seemingly incomplete set of linear measurements. First, we shall briefly present a few recovery algorithms: orthogonal matching pursuit, iterative hard thresholding, $\ell_1$-minimization. We will then concentrate on the measurement process, where randomness plays the major part since it allows to use the asymptotically minimal number of measurements. The classical analysis involves matrices obeying a restricted isometry property with respect to the $\ell_2$-norm --- typically subgaussian matrices --- but we will also investigate other random matrices, such as matrices with independent subexponential entries or adjacency matrices of lossless expanders. We will further discuss some nonuniform recovery results about structured random matrices, when an additional randomness on the signals under consideration is introduced.

16h00 Matthieu Kowalski (Paris-Sud 11, L2S)

Parcimonie, structures et normes mixtes.

Résumé: Une approche variationnelle classique pour aborder les problèmes inverses mal posés, est l'utilisation d'une attache aux données l2 régularisée par une norme bien choisie sur des coefficients de synthèse. La norme l1, connue et utilisée pour obtenir des décompositions parciminieuses, considère les coefficients comme i.i.d., hypothèse parfois trop forte. On présente l'utilité (mais aussi les limites) des normes mixtes qui permettent de structurer ces coefficients, et ainsi d'exploiter des a priori plus fort que la parcimonie. Cette approche est appliquée à divers problèmes comme la décomposition de signaux en "couches" et la séparation/localisation de sources.