Séminaire PAS

Juin 2025

• Jeudi 12 juin 2025 - Sebastien Bossu
  

Mai 2025

• Jeudi 22 mai 2025 - Tom Carroll (University College Cork)

  An uncertainty principle for the Vaserstein distance

  This is joint work with Xavier Massaneda and Joaquim Ortega-Cerdà (Barcelona). I will discuss an uncertainty principle of the following form: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be large. The result in two dimensions is due to Steinerberger. A partial result in higher dimensions, which we improve upon, is due to Sagiv and Steinerberger. Related to this is a sharp upper estimate of the cost of transporting the positive part of an eigenfunction of the Laplacian. This proves a conjecture of Steinerberger and provides a lower bound on the size of a nodal set of eigenfunctions.

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• Jeudi 15 mai 2025 - Victor-Emmanuel Brunel

  Deep points in curved spaces

  In this talk, based on a joint work with Shin-ichi Ohta (Osaka University) and Jordan Serres (Institut National des Sciences Appliquées, Toulouse), I will present a generalization of Grunbaum’s inequality to curved spaces. This inequality states that in Euclidean spaces, uniform distributions on convex bodies and, more generally, log-concave distributions, have deep points in Tukey’s sense: There always exists a point such that any halfspace containing that point has mass bounded from below by some universal positive constant. This fact is important in high dimensional statistics because it allows to discriminate deep points from shallow points independently of the ambient dimension.
  
  I will briefly introduce RCD spaces, which are Riemannian-like spaces that satisfy a curvature-dimension condition, similar to that of Bakry and Emery, which is tied to measure concentration and fast mixing of Markov processes on Riemannian manifolds. These spaces offer a natural framework in which splitting theorems allow to extend the definition of halfspaces and, as in Euclidean spaces, to reduce the proof of Grunbaum’s inequality to one-dimensional computations.

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Avril 2025

• Jeudi 17 avril 2025 - Sana Louhichi

  Bounds on the Hausdorff Distance for Stationary, Compactly Supported Sequences with Applications to Topological Reconstruction

  In topological data analysis, controlling the Hausdorff distance plays a crucial rule in persistent homology and shape reconstruction.
  Throughout this talk, we consider a sequence of stationary, weakly dependent, compactly supported random variables.
  We first propose different methods to bound the Hausdorff distance from this stationary random sequence to its common support.
  The obtained bounds allow us to achieve the optimal rate of convergence for the i.i.d. case (proved by Chazal et al. in 2015).
  We discuss different types of dependence for the considered stationary sequence.
  Next, we present a novel topological reconstruction result and provide some illustrative examples.
  This talk is based on joint work with Sadok Kallel.

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• Lundi 07 avril 2025 - Gunnar Taraldsen

  Conformal prediction distributions

  Confidence distributions can be seen as a frequentist alternative to Bayesian posteriors for summarizing the knowledge available for an unknown quantity based on the observed data and a model, but without the need for a Bayesian prior. Two equally important parts of a confidence distribution are (i) A distribution estimator, ii) A matching family of confidence-credibility regions. This is illustrated by examples for two-dimensional parameters, but also for one-dimensional parameters. The main tools for construction are given by a) Data generating algorithms, b) Pivots, and c) p-value functions. Presently, confidence distributions are in practice not anywhere near the success of Bayesian posteriors due to a lack of general available software and a lack of a fully developed theory. This can and will be changed as part of the focus in machine learning is directed towards uncertainty quantification in its predictions. It will be explained that a prediction distribution generalizes the concept of a confidence distribution by being defined by two equally important parts: (i) A distribution estimator, ii) A matching family of prediction-credibility regions. The methods can also be used to construct multivariate conformal prediction distributions of use in an on-line setting in which multivariate labels are predicted successively, each one being revealed before the next is predicted. The corresponding sequence of families of conformal prediction regions are characterized by successive predictions being right a fraction of the time given by the corresponding credibility level even though they are based on an accumulating data set rather than on independent data sets. Optimality is discussed in terms of frequentist properties of the distribution estimator and of the family of credibility regions separately. Uniformly most powerful unbiased distributions and regions are obtained for exponential families and group families in concrete cases. More generally, Bayes optimal regions are obtained using Bayes informed frequentist methods.
  
  Keywords: Conformal prediction; sufficient statistics; equivariance; disintegration; hypothesis testing; Neyman-Pearson lemma; confidence region

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• Jeudi 03 avril 2025 - Gunnar Taraldsen

  Improper surprises in probability and statistics

  Improper distributions are routinely used as default priors in Bayesian computations. Most often this is done without reference to a proper theory. Lindley (1965) adopted the theory of Rényi (1955) to include improper distributions but later banned the use of improper priors after seeing the surprising marginalization paradoxes of Dawid, Stone and Zidek (1973). This surprise, and many others, are explained if the consequences of the theory of Rényi are analyzed more closely. A purely probabilistic example is given by Brownian motion: The stationary distribution is the uniform, and it is obtained as a limiting distribution using q-vague convergence introduced by Bioche and Druilhet (2016). An improper distribution can also appear as the posterior from a proper prior. It is explained that this is a consequence of defining a unique posterior by choosing a continuous version. The main mathematical foundation is given by the constructive proof of existence of a transformation from prior to posterior knowledge defined by Taraldsen, Tufto and Lindqvist (2022). The posterior always exists and is uniquely defined by the prior, the observed data, and the statistical model. The transformation is, as it should be, an extension of conventional Bayesian inference as defined by the axioms of Kolmogorov. It is an extension since the novel construction is valid also when replacing the axioms of Kolmogorov by the axioms of Rényi for a conditional probability space.
  
  
  
  Keywords: axioms of probability and statistics, Bayesian statistics, conditional law, Gibbs sampling, intrinsic Gaussian Markov random fields, marginalization paradox, Jeffreys-Lindley paradox, q-vague convergence
  
  
  
  Note: The presentation is based on joint work with Bo Henry Lindqvist and Jarle Tufto.

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Mars 2025

• Jeudi 27 mars 2025 - Marjolaine Puel (attention horaire modifié : 11h15)

  Diffusion fractionnaire comme approximation de modèles cinétiques.

  Dans cet exposé, nous expliquerons tout d’abord comment approcher la solution d’une équation cinétique représentant au niveau mésoscopique le mouvement de particules qui se cognent par un profile donné en vitesse multiplié par une densité solution d’une équation de diffusion. Nous montrerons ensuite pourquoi, lorsque les équilibres sont à queue lourde, c’est à dire que les particules à très grande vitesse ont une densité non négligeable, la description macroscopique fait intervenir un phénomène de saut. Enfin, nous introduirons une méthode spectrale qui permet de capter les différents seuils dans les valeurs des paramètres déterminant l’équation limite adéquate.

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• Jeudi 13 mars 2025 - Corentin Correia (Institut de Mathématiques de Jussieu–Paris Rive Gauche)

  Odomutants and flexibility results for quantitative orbit equivalence

  Two measurable bijections of a standard probability space are orbit equivalent if they have the same orbits up to conjugacy. In recent years, odometers have been a central class of systems for explicit constructions of orbit equivalences, using their combinatorial structure. In this talk we introduce a construction of orbit equivalence between odometers and new systems that we call odomutants. The starting point for this notion is a construction of Feldman in 1976, which enables us to get a first flexibility result about even Kakutani equivalence. Here we deal with a second result, about entropy.
  It follows from work of Kerr and Li that if the cocycles are log integrable, the entropy is preserved. Our construction of odomutants shows that their result is optimal, namely we find odomutants of positive entropy orbit equivalent to an odometer, with almost log integrable cocycles.

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Février 2025

• Jeudi 06 février 2025 - Eddie Aamari

  A theory of stratification learning: clustering-by-dimensionality with reconstruction

  Given i.i.d. sample from a stratified mixture of immersed manifolds of different dimensions, we will study the minimax estimation of the underlying stratified structure. We will provide a constructive algorithm allowing to estimate each mixture component at its optimal dimension-specific rate adaptively. The method is based on an ascending hierarchical co-detection of points belonging to different layers, which also identifies the number of layers and their dimensions, assigns each data point to a layer accurately, and estimates tangent spaces optimally. These results hold regardless of any ambient assumption on the manifolds or on their intersection configurations. They open the way to a broad clustering framework, where each mixture component models a cluster emanating from a specific nonlinear correlation phenomenon.

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Janvier 2025

• Jeudi 16 janvier 2025 - Étienne Matheron (Université d'Artois)

  Opérateurs héréditairement fréquemment hypercycliques

  

Décembre 2024

• Jeudi 05 décembre 2024 - Arthur Stephanovic

  Smooth transport map via diffusion process and applications to generative modelling

  We prove that the Langevin map transporting the d-dimensional
  Gaussian to a k-smooth déformation is (k+1)-smooth. We give applications of this result to functional inequalities as well as generative modelling.

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Novembre 2024

• Jeudi 21 novembre 2024 - Jordan Serre

  Polchinski Renormalization Group, dynamical Gamma-calculus and transportation of measures.

  Renormalization is a set of techniques developed by theoretical physicists to deal with the appearance of divergent quantities in quantum field theory. Highly controversial because they were not mathematically rigorous, these techniques have become increasingly accepted since the work of K. Wilson in the 1970s, showing how they can be used to understand phase transition phenomena in statistical physics, followed by the famous article by J. Polchinski in 1984 on effective Lagrangians.
  Although renormalization is now well accepted and understood in physics, it remains a challenge to provide the appropriate mathematical theory that explains it.
  In this talk, we will focus on the approach of R. Bauerschmidt and T. Bodineau. We will try to explain it through the continuum phi-4 model, then show how certain quantities such as mixing times can be studied along the Renormalization flow, and finally how this flow can be used to construct Lipschitz transport maps.

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• Jeudi 14 novembre 2024 - Valentin Gillet (Université de Lille)

  Propriétés typiques de contractions positives et problème du sous-espace invariant

  Le problème du sous-espace invariant pour les opérateurs positifs, toujours ouvert à ce jour, s'énonce ainsi : étant donné un espace de Banach X complexe de dimension infinie et ayant une base, est-ce que tout opérateur positif sur X admet un sous-espace invariant non trivial ? L'un des résultats fondamentaux à ce sujet, dû à Abramovich, Aliprantis et Burkinshaw, montre l'existence d'un sous-espace invariant non trivial pour tout opérateur positif sur un espace de Banach X ayant une base qui commute avec un opérateur positif non nul et quasinilpotent en un certain vecteur positif non nul.
  
  Dans cet exposé, nous déterminerons, pour les topologies SOT et SOT*, si l'ensemble des contractions positives sur X vérifiant les hypothèses du théorème d'Abramovich, Aliprantis et Burkinshaw est une partie comaigre dans l'ensemble des contractions positives.
  

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Octobre 2024

• Jeudi 03 octobre 2024 - Harouna Kinda

  Natural Resource Revenues and Double Taxation Treaties in Developing Countries: Evidence from A Neural Network Approach

  This paper investigates the effects of double taxation treaties on resource revenue mobilization in 83 resource-rich countries from 2000 to 2019 by applying standard panel fixed effects and methods-of-moments approaches. We calculate countries’ centrality indices by year based on their importance in the tax treaty network and show that centrality indices have a negative relationship with resource revenue mobilization—findings that are robust to alternative centrality indices and government revenue aggregates. We also use the betweenness centrality index to identify countries characterized as intermediate jurisdictions (countries -classified as investment or tax hubs based on their betweenness centrality index, which is above the median), arguing that multinational companies structure their investments to benefit from the low withholding tax rates in these countries. Applying the entropy balancing method, we find evidence of a negative effect on resource revenue mobilization dueto signing tax treaties with country-classified investment or tax hubs

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Septembre 2024

• Jeudi 26 septembre 2024 - Wiliam Kengne

  Robust deep learning from weakly dependent data

  We consider robust deep learning from weakly dependent observations, with unbounded loss function and unbounded input/output. It is only assumed that the output variable has a finite r order moment, with r >1.
  Non asymptotic bounds for the expected excess risk of the deep neural network estimator are established under strong mixing assumptions on the observations.
  We derive a relationship between these bounds and r, and when the data have moments of any order (that is r=\infty), the convergence rate is close to some well-known results.
  When the target predictor belongs to the class of H\"older smooth functions with sufficiently large smoothness index, the rate of the expected excess risk for exponentially strongly mixing data is close to that obtained with i.i.d. samples.
  Application to robust nonparametric regression with heavy-tailed errors shows that, robust estimators with absolute loss and Huber loss outperform the least squares method.

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