Séminaire de mathématiques appliquées (archives)

Many sources of our informational landscape can be formalized as a network of intertwined documents and authors. For a long time the textual content of documents and the structure that shows how documents and authors relate to each other have been considered separately. Recently document network embedding has been proposed to learn representations that take both content and structure into account. This space can then be used for downstreams tasks, such as classification or link prediction. In this talk I will give an overview of recent methods that aim at building such embedding spaces. In particular, I will focus on several models that were recently proposed in the ERIC Lab [1,2,3,4].

[1] R. Brochier, A. Guille and J. Velcin. Global Vectors for Node Representation. Proceedings of The Web Conference (WWW), 2019. [2] A. Gourru, J. Velcin, J. Jacques and A. Guille. Document Network Projection in Pretrained Word Embedding Space, Proceedings of ECIR, 2020. [3] R. Brochier, A. Guille and J. Velcin. Inductive Document Network Embedding with Topic-Word Attention, Proceedings of ECIR, 2020. [4] A. Gourru, J. Velcin and J. Jacques. Gaussian Embedding of Linked Documents from a Pretrained Semantic Space, Proceedings of IJCAI, 2020.

Je présenterai des travaux sur le mouvement brownien réfléchi en dimension 2. Introduite dans les années 1980 notamment par Harrison, Varadhan, Williams, cette diffusion est au cœur de nombreux problèmes probabilistes. Malgré tout l'intérêt que ce processus aléatoire a suscité pendant ces quarante dernières années, peu de résultats existaient sur sa distribution d'équilibre (lorsqu'elle existe), à l'exception de quelques cas où, de façon remarquable, la densité associée est une somme d'exponentielles, comme la fameuse skew symmetry. Dans cet exposé, je m'intéresserai à caractériser géométriquement les paramètres du modèle pour lesquels la distribution stationnaire est plus simple qu'attendu (en un certain sens que je préciserai). Dans ces cas, nous produisons une formule effective pour la densité. D'un point de vue technique, notre approche mélange calcul stochastique et équations fonctionnelles, en particulier les équations aux q-différences. Il s'agit d'un travail commun avec Mireille Bousquet-Mélou, Andrew Elvey Price, Sandro Franceschi et Charlotte Hardouin ; le preprint se trouve à l'adresse https://arxiv.org/abs/2101.01562.

Pour illustrer la sélection d'individus les plus adaptés à un environnement donné à partir d'un modèle de population structurée par une variable de trait, on peut étudier la convergence de la distribution de population vers une masse de Dirac concentrée en ce trait adapté. Dans cet exposé, je présenterai des résultats sur le comportement asymptotique de la solution d'une équation structurée en âge et en trait. Dans un premier temps, j'introduirai un modèle simplifié en supposant qu'il n'y pas de mutation. L'analyse de ce modèle repose sur l'étude d'un problème aux valeurs propres paramétré par la variable de trait. Ensuite, je présenterai le modèle avec mutations qui fait apparaître une équation de Hamilton-Jacobi sous contraintes. Il s’agit d’un travail fait en collaboration avec Samuel Nordmann et Benoît Perthame

Many spatio-temporal data record the time of birth and death of individuals, along with their spatial trajectories during their lifetime, whether through continuous-time observations or discrete-time observations. Natural applications include epidemiology, individual-based modelling in ecology, spatio-temporal dynamics observed in bio-imaging, and computer vision. To model this kind of data, we introduce spatial birth-death-move processes, where the birth and death dynamics depends on the current spatial state of all alive individuals and where individuals can move during their lifetime according to a continuous Markov process. We present some of the basic probabilistic properties of these processes and we consider the non-parametric estimation of their birth and death intensity functions. The setting is original because each observation in time belongs to a non-vectorial, infinite dimensional space and the dependence between observations is barely tractable. We prove the consistency of kernel estimators in presence of continuous-time or discrete-time observations, under fairly simple conditions. We moreover discuss how we can take advantage in practice of structural assumptions made on the intensity functions and we explain how data-driven bandwidth selection can be conducted, despite the unknown (and sometimes undefined) second order moments of the estimators. We finally apply our statistical method to the analysis of the spatio-temporal dynamics of proteins involved in exocytosis in cells.

This is a joint work with Ronan Le Guével (Rennes 2).

Joint models for longitudinal and survival data are now widely used in biostatistics to address a variety of etiological and predictive questions. Originally designed to simultaneously analyze the trajectory of a single marker measured repeatedly over time, and the risk of a single right-censored time-to-event, joint models now need to capture increasingly complex longitudinal information to adapt to in-depth medical research questions. After an introduction of the general joint modelling methodology, I discuss two extensions proposed to handle multivariate repeated marker information: one using a latent class approach and one using a latent degradation process approach. The models are illustrated on real data, in particular to describe the progression of Multi-System Atrophy, a rare neurodegenerative disease.

Variational discretizations are known for preserving key physical invariants in a natural way, leading to long-time stability properties. In this talk I will present a discrete action principle for the Vlasov-Maxwell equations that applies in a general structure-preserving discrete framework.

In this framework the finite-dimensional electromagnetic potentials and fields are represented in a discrete de Rham sequence involving general Finite Element spaces, and the particle-field coupling is represented by a set of projection operators that commute with the differential operators.

One application of this approach is a new variational spectral PIC method that has a discrete Hamiltonian structure and relies on particle-field coupling techniques very similar to those encountered in standard PIC schemes.

This is a joint work with Jakob Ameres, Katharina Kormann and Eric Sonnendrücker from the Max Planck IPP in Garching, Germany

Classical approaches to multiple testing grant control over the amount of false positives for a specific method prescribing the set of rejected hypotheses. In practice many users tend to deviate from a strictly prescribed multiple testing method and follow ad-hoc rejection rules, tune some parameters by hand, compare several methods and pick from their results the one that suits them best, etc. This will invalidate standard statistical guarantees because of the selection effect. To compensate for any form of such ”data snooping”, an approach which has garnered significant interest recently is to derive ”user-agnostic”, or post hoc, bounds on the false positives valid uniformly over all possible rejection sets; this allows arbitrary data snooping from the user. We present two contributions: starting from a common approach to post hoc bounds taking into account the p-value level sets for any candidate rejection set, we analyze how to calibrate the bound under different assumptions concerning the distribution of p-values. We then build towards a general approach to the problem using a family of candidate rejection subsets (call this a reference family) together with associated bounds on the number of false positives they contain, the latter holding uniformly over the family. It is then possible to interpolate from this reference family to find a bound valid for any candidate rejection subset. This general program encompasses in particular the p-value level sets considered earlier; we illustrate its interest in a different context where the reference subsets are fixed and spatially structured. [Joint work with Pierre Neuvial and Etienne Roquain]

Considering a Poisson process observed on a bounded, fixed interval, we are interested in the problem of detecting an abrupt change in its distribution, characterized by a jump in its intensity. Formulated as an off-line change-point problem, we address two distinct questions : the one of detecting a change-point and the one of estimating the jump location of such change-point once detected. This study aims at proposing a non-asymptotic minimax testing set-up, first to construct a minimax and adaptive detection procedure and then to give a minimax study of a multiple testing procedure designed for change-point localisation.

Mathieu Ribatet vous invite à une réunion Zoom planifiée.

Sujet : Séminaire de Fabrice Grela Heure : 9 févr. 2021 11:00 AM Paris

Participer à la réunion Zoom https://ec-nantes.zoom.us/j/99612428063

https://www.lebesgue.fr/fr/content/seminars-jrna2021

We study the approximation of multivariate functions on bounded domains with tensor networks (TNs). The main conclusion of this work is an answer to the following two questions that can be seen as different perspectives on the same issue: “What are the approximation capabilities of TNs?” and “What is the mathematical structure of approximation classes of TNs?”

To answer the former: we show that TNs can (near to) optimally replicate h-uniform, h-adaptive and hp-approximation. Tensor networks thus exhibit universal expressivity w.r.t. classical polynomial-based approximation methods that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to optimally approximate many function classes – without being adapted to the particular class in question.

To answer the latter: we show that approximation classes of TNs are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into TN approximation classes and that TN approximation classes themselves are not embedded in any classical smoothness space. Although approximation with TNs is a highly nonlinear approximation scheme, under certain restrictions on the topology of the network, the set of functions that can be approximated with TNs at a given rate is highly structured – it is a (quasi-)Banach space. Moreover, it is a very “large” space: many well-known classical smoothness spaces, such as isotropic/anisotropic/mixed Besov spaces, are embedded therein – while it also contains exotic functions that have no smoothness in the classical sense.

Mathieu Ribatet vous invite à une réunion Zoom planifiée.

Sujet : Séminaire Ali Mazen Heure : 26 janv. 2021 11:00 AM Paris

Participer à la réunion Zoom https://ec-nantes.zoom.us/j/94571681620