Probabilities and Statistic seminar

We define a wide class of Markovian load balancing queueing networks, including classic networks studied in the lively literature on the subject. Each network has identical single-server infinite-buffer queues and implements a load balancing policy to allocate each task at its arrival and possibly reallocate it at service completions. The purpose of the policy is to optimize server utilization under constraints such as limited information, real-time decision taking, and network topology. The queue length process is not necessarily exchangeable. The invariant law is in general not known even up to normalizing constant. We provide perfect simulation methods in view of Monte Carlo estimation of quantities of interest in equilibrium, for instance for performance evaluation. In this infinite multi-dimensional state space, we use an unusual preorder defining an order up to permutation of the coordinates, define a coupling in which networks in this class are dominated by the network with uniform routing, and implement dominated coupling from the past methods.

[The talk will be in French, but slides will be in English.]

(Exceptionnellement, le séminaire aura lieu à Metz et sera diffusé en visio en salle de conférence à Nancy.)

Planning, titres et résumés ici.

Chemical reactions network inside cells have been extensively studied in order to better understand various biological phenomena. The majority of experimental studies are performed with cells that are part of a growing population. This population context is rarely taken into account even if selection between cells (due for example to growth) takes places within the studied system.
In this talk, I will represent such systems as continuous-time Markov chains. The measure-valued Markov process of the cell population will take into account the chemical reactions inside the cells as well as reactions between cells. By conditioning on non-absorption, we derive an equation for the expected population distribution within a growing population.
This extension of the Chemical Master Equation provides us a new framework to study cell population dynamics. I will present theoretical results on long-term behaviour of the population (stationary distribution, growth rate of the population) and an application of this framework to experimental data.

The increasing availability of temporal and geo-coded data underscores the importance of spatio-temporal statistical modelling in tackling complex issues across various real-world settings. In the first part of this talk, we will briefly showcase novel spatio-temporal statistical methods developed to model various types of data defined both in space and time (e.g., time-series, point patterns, lattice data, geostatistical data, etc.), with a focus on applications in environmental and public health domains. In the second part, we will (I) delve into the modelling of trajectory (or path) data and (II) explore the details of a statistical method for addressing spatially varying preferential sampling when modelling geostatistical data. Specifically, we will account for preferential sampling by including a spatially varying coefficient that describes the dependence strength between the process that models the sampling locations and the corresponding latent field. We achieve this by approximating the preferentiality component with a set of basis functions, with the corresponding coefficients estimated using the integrated nested Laplace approximation (INLA) method. This approach allows for efficient inference with a low computation burden.

The push-forward operation enables one to redistribute a probability measure through a deterministic map. It plays a key role in statistics and optimization: many learning problems (notably from optimal transport, generative modeling, and algorithmic fairness) include constraints or penalties framed as push-forward conditions on the model. However, the literature lacks general theoretical insights on the (non)convexity of such constraints and its consequences on the associated learning problems. The presented work aims at filling this gap. In a first part, we provide a range of sufficient and necessary conditions for the (non)convexity of two sets of functions: the maps transporting one probability measure to another; the maps inducing equal output distributions across distinct probability measures. This highlights that for most probability measures, these push-forward constraints are not convex. In a second time, we show how this result implies critical limitations on the design of convex optimization problems for learning generative models or group-fair predictors. This work will hopefully help researchers and practitioners have a better understanding of the critical impact of push-forward conditions onto convexity.

On s’intéresse à l’étude de couplages des mouvements browniens sous-elliptiques sur plusieurs variétés sous-riemaniennes: les groupes de Carnot libres d’ordre 2, incluant le groupe d’Heisenberg, ainsi que les groupes de matrices $SU(2)$ et $SL(2,\mathbb{R})$. Après une rapide introduction aux structures sous-Riemannienne, nous proposerons plusieurs méthodes explicites de couplages markoviens ou non markoviens. En particulier ces constructions mènent à des estimées du taux de couplage dont on déduit des inégalités pour le semi-groupe de la chaleur et pour les fonctions harmoniques que nous expliciterons.

Pour finir nous présenterons un nouveau modèle de couplage non markovien “en un coup” sur tous les groupes de Carnot libres de profondeur 2. Il permet notamment d’obtenir des relations similaires à la formule de Bismut-Elworthy-Li pour les gradients de semi-groupes via l’étude d’un changement de probabilité sur l’espace des vecteurs Gaussiens.

On considère un problème de contrôle consistant à trouver une trajectoire reliant un point initial x à un point cible y, le système se déplaçant uniquement dans certaines directions admissibles. On suppose que les champs de vecteurs correspondants satisfont la condition de Hörmander, de telle sorte que par un théorème classique (Chow-Rashevskii), il existe des trajectoires qui satisfont cette contrainte. Une manière naturelle d’essayer de résoudre ce problème est via un flot de gradient sur l’espace des contrôles. Cependant, la dynamique correspondante peut avoir des point-selles, et pour obtenir un résultat de convergence il faut donc faire des hypothèses (par exemple probabilistes) sur la condition initiale. Dans ce travail, nous considérons le cas où cette initialisation est irrégulière, que nous formulons grâce à la théorie des trajectoires rugueuses de Lyons. Dans des cas simples, on prouve que le flot de gradient converge vers une solution, si la condition initiale est une trajectoire d’un mouvement Brownien (ou d’un processus de régularité plus faible). La preuve combine des idées de calcul de Malliavin avec des inégalités de Łojasiewicz. Une motivation possible pour nos travaux vient de l’entraînement de réseaux de neurones résiduels profonds, dans un régime où le nombre de paramètres par couche est fixé, et la dimension du vecteur de données est élevée. Il s’agit d’un travail en collaboration avec Florin Suciu (Paris Dauphine).