Séminaire de mathématiques appliquées (archives)

The statistical query (SQ) framework consists in replacing the usual access to samples from a distribution, by the access to adversarially perturbed expected values of functions interactively chosen by the learner. This framework provides a natural estimation complexity measure, enforces robustness through adversariality, and is closely related to differential privacy. In this talk, we study the SQ complexity of estimating d-dimensional submanifolds in R^n. We propose a purely geometric algorithm called Manifold Propagation, that reduces the problem to three local geometric routines: projection, tangent space estimation, and point detection. We then provide constructions of these geometric routines in the SQ framework.

The aim of this work is to propose a numerical method to solve parameter-dependent hyperbolic partial differential equations (PDEs) with a moment approach, based on a previous work from Marx et al. (2020). This approach relies on a very weak notion of solution of nonlinear equations, namely parametric entropy measure-valued (MV) solutions, satisfying linear equations in the space of Borel measures. The infinite-dimensional linear problem is approximated by a hierarchy of convex, finite-dimensional, semidefinite programming problems, called Lasserre's hierarchy. This gives us a sequence of approximations of the moments of the occupation measure associated with the parametric entropy MV solution, which is proved to converge.

The aim of this work is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson partial differential equation (PDE) with Neumann boundary condition. Using Barron’s representation of the solution with a probability measure defined on the set of parameter values, the energy is minimized thanks to a gradient curve dynamic on the 2-Wasserstein space of the set of parameter values defining the neural network. Inspired by a work from Bach and Chizat, we prove that if the gradient curve converges, then the represented function is the solution of the elliptic equation considered. In contrast to previous works, the activation function we use here is not assumed to be homogeneous to obtain global convergence of the flow.

Adipose cells or adipocytes are the specialized cells composing the adipose tissue in a variety of species.Their role is the storage of energy in the form of a lipid droplet inside their membrane. Based on the amount of lipid they contain, one can consider the distribution of adipocyte per amount of lipid and observe a peculiar feature : the resulting distribution is bimodal, thus having two local maxima. The aim of this talk is to introduce a model built from the work in Soula & al. (2013) that is able to reproduce this bimodal feature using a Lifshitz-Slyozov model. Additionally we present some result on this model and its relation to the Becker-Döring model.

Dans cet exposé, nous présentons des modèles asymptotiques obtenus à l'aide de développements multi-échelles qui permettent de décrire des phénomènes électromagnétiques ou géophysiques. Dans une première partie, nous présentons des modèles d’impédance pour la résolution de problèmes de couche limite ou de couche mince. La précision et la performance des modèles obtenus sont illustrées par des résultats numériques. Dans une seconde partie, nous présentons une méthode de paramétrisation du potentiel magnétique pour un problème de courant de Foucault dans des matériaux ferromagnétiques.

We study the approximation of a square integrable function from a finite number of its evaluations at well-chosen nodes. The function is assumed to belong to a reproducing kernel Hilbert space (RKHS), the approximation to be one of a few natural finite-dimensional approximations, and the nodes to be drawn from one of two probability distributions. Both distributions are related to determinantal point processes, and use the kernel of the RKHS to favor RKHS-adapted regularity in the random design. While previous work on determinantal sampling relied on the RKHS norm, we prove mean-square guarantees in L2 norm.

Nous étudions un problème d'évolution basique qui est l'invasion d'un mutant (variant) dans une population résidente. Pour des modèles simples nous donnerons quelques outils mathématiques qui permettent de donner une réponse à la question de l'invasion dans les environnements constants, périodiques et aléatoires.

Many functions of interest exhibit weighted summability of the coefficients with respect to some dictionary of basis functions. This property allows us to derive close-to-optimal best n-term approximation rates by means of a new weighted version of Stechkin’s Lemma. It is well known that these best n-term approximations can be estimated efficiently from samples. In fact, highly refined sample complexity results guarantee quasi-best n-term approximations with high probability. But the algorithm that perform this approximation have a time complexity that is linear in the size of the dictionary of basis functions. This is problematic, since this size typically grows rapidly when the dimension of the sought function increases.

La plupart des processus environnementaux (intempéries, température, vent...) sont spatiaux par nature et la modélisation de leurs extrêmes nécessite de prendre en compte cette caractéristique. Souvent, on peut également rattacher à ces processus extrêmes environnementaux un autre process, à valeur angulaire (rafales de vent et leurs directions, forte intempéries et leur jour d’occurrence...). Dans cette présentation, je présenterai une nouvelle approche statistique pour la modélisation jointe de processus spatiaux extrêmes et angulaire, ainsi que des résultats et applications sur l’efficacité de cette approche.

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Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric modelling of the population densities. Identifiability fails when data are in fact independent, and we study the frontier between learnable and unlearnable two-state nonparametric HMMs. Interesting new phenomena emerge when the cluster distributions are modelled via density functions (the `emission' densities) belonging to standard smoothness classes compared to the multinomial setting [1]. Notably, in contrast to the multinomial setting previously considered, the identification of a direction separating the two emission densities becomes a critical, and challenging, issue.