Résumés

jeudi 22 septembre 2022

Frédéric Chazal (LMO, INRIA Saclay)

""Comprendre la structure topologique des données : une introduction à l'homologie persistante"

Bruno Galerne (IDP, Université d'Orléans)

"Champs aléatoires pour la synthèse de textures"

vendredi 23 septembre 2022

Simon Barthelmé (GIPSA, CNRS/Université Grenoble Alpes)

"Kernel matrices in the flat limit"

Kernel matrices are ubiquitous in statistics and machine learning. Within Bayesian statistics they occur most often as covariance matrices of Gaussian processes, in non-parametric or semi-parametric models. Most of the theoretical work on kernel methods has focused on a large-$n$ asymptotics, characterising the behaviour of kernel matrices as the amount of data increases. Fixed-sample analysis is much more difficult outside of simple cases, such as locations on a regular grid.

In this talk I will describe a fixed-sample analysis that was first studied in the context of approximation theory by Fornberg \& Driscoll (2002), called the ``flat limit''. In flat-limit asymptotics, the goal is to characterise kernel methods as the length-scale of the kernel function tends to infinity, so that kernels appear flat over the range of the data. Even though flat kernel matrices may seem trivial, because their rank goes to one, detailed analysis reveals very interesting structure. We have been able to show that the eigenvectors and eigenvalues in that regime are tightly related to orthogonal polynomials or splines, depending on the smoothness of the kernel.
I will introduce some applications to Determinantal Point Processes and Gaussian processes.

This is joint work with K. Usevich, N. Tremblay, P.-O. Amblard

Bartek Blaszczyszyn (Dyogene, INRIA Paris)

"Particle gradient descent model for point process generation"

We introduces a generative model for planar point processes in a square window, built upon a single realization of a stationary, ergodic point process observed in this window. Inspired by recent advances in gradient descent methods for maximum entropy models, we propose a method to generate similar point patterns by jointly moving particles of an initial Poisson configuration towards a target counting measure. The target measure is generated via a deterministic gradient descent algorithm, so as to match a set of statistics of the given, observed realization. Our statistics are estimators of the multi-scale wavelet phase harmonic covariance, recently proposed in image modeling. They allow one to capture geometric structures through multi-scale interactions between wavelet coefficients. Both our statistics and the gradient descent algorithm scale better with the number of observed points than the classical K-nearest neighbour distances previously used in generative models for point processes, based on the rejection sampling or simulated-annealing. The overall quality of our model is evaluated on point processes with various geometric structures through spectral and topological data analysis.

Joint work with Antoine Brochard, Stéphane Mallat and  Sixin Zhang
https://arxiv.org/abs/2010.14928
to appear in Statistics and Computing, Springer 2022+

Claire Brécheteau (LMJL, Centrale Nantes)

"Approximating data with a union of ellipsoids and clustering"

Agnès Desolneux (Centre Borelli, CNRS/ENS Paris-Saclay)

"Les processus ponctuels déterminantaux à l'intersection de la géométrie stochastique et du traitement d'image"

Dans ce travail, effectué en collaboration avec Bruno Galerne (Université d'Orléans) et Claire Launay (Albert Einstein College of Medicine, New-York), nous nous intéressons à la définition et à l'utilisation de processus ponctuels déterminantaux (qui sont des processus répulsifs) pour deux tâches en traitement d'image : d'une part la synthèse de texture par modèle de type shot-noise, et d'autre part le sous-échantillonnage de l'ensemble des patchs d'une image.

Nicolas Keriven (GIPSA, CNRS/Université Grenoble Alpes)

"Graph Neural Networks on Large Random Graphs: Convergence, Stability, Universality"

In this talk, we will discuss some theoretical properties of Graph Neural Networks (GNNs) on large graphs. Indeed, most existing analyses of GNNs are purely combinatorial and may fail to reflect their behavior regarding large-scale structures in graphs, such as communities of well-connected nodes or manifold structures. To address this, we assume that the graphs of interest are generated with classical models of random graphs. We first give non-asymptotic convergence bounds of GNNs toward "continuous'' equivalents as the number of nodes grows. We then study their stability to small deformations of the underlying random graph model, a crucial property in traditional CNNs. Finally, we study their universality and approximation power, and show how some recent GNNs are more powerful than others. This is a joint work with Samuel Vaiter (CNRS) and Alberto Bietti (NYU).

Contributions

Jules Mabon (INRIA Sophia-Antipolis)

"Processus ponctuels marqués et CNNs pour la détection d'objets dans des images de télédétection"