Prochainement

Jeudi 30 mars 14:00-15:00 Bastien Mallein (DMA (ENS Paris))
Processus de branchement infiniment divisible

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iCal

Notes de dernières minutes : Un processus de branchement infiniment divisible est un processus à temps continu dont tous les squelettes discrets sont des marches aléatoires branchantes. On montre qu’un tel processus est un processus de Lévy branchant : un processus de particule produisant des enfants selon une dynamique de Poisson. Ce résultat est l’analogue (pour les processus de branchement) de l’équivalence entre processus de Lévy et variables aléatoires infiniment divisibles.

Processus de branchement infiniment divisible  Version PDF

Jeudi 20 avril 14:00-15:00 Irène Marcovici  (IECL (Université de Lorraine))
Ergodicité de certains automates cellulaires bruités

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iCal

Notes de dernières minutes : Quand on perturbe un automate cellulaire par un bruit aléatoire (probabilité positive d’erreur, indépendamment pour différentes cellules), on s’attend généralement à ce que le système soit ergodique, c’est-à-dire à ce qu’il oublie progressivement la configuration initiale au cours de son évolution. Lorsque le bruit est suffisamment élevé, des méthodes classiques de couplage permettent de le montrer. Mais lorsque le bruit est faible, l’ergodicité est souvent difficile à prouver. Je présenterai différentes extensions de la méthode de couplage lorsque l’automate cellulaire a des propriétés spécifiques (nilpotence, permutivité...).

Ergodicité de certains automates cellulaires bruités  Version PDF

Passés

Jeudi 23 mars 14:00-15:00 Julien Stoehr  (Insight)
Hidden Gibbs random fields model selection using Block Likelihood Information Criterion

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Résumé : Performing model selection between Gibbs random fields is a very challenging task. Indeed, because of the Markovian dependence structure, the normalizing constant of the fields cannot be computed using standard analytical or numerical methods. Furthermore, such unobserved fields cannot be integrated out, and the likelihood evaluation is a doubly intractable problem. This forms a central issue to pick the model that best fits an observed data. We introduce a new approximate version of the Bayesian Information Criterion. We partition the lattice into contiguous rectangular blocks, and we approximate the probability measure of the hidden Gibbs field by the product of some Gibbs distributions over the blocks. On that basis, we estimate the likelihood and derive the Block Likelihood Information Criterion (BLIC) that answers model choice questions such as the selection of the dependence structure or the number of latent states. We study the performances of BLIC for those questions. In addition, we present a comparison with ABC algorithms to point out that the novel criterion offers a better trade-off between time efficiency and reliable results.

Hidden Gibbs random fields model selection using Block Likelihood Information Criterion  Version PDF

Jeudi 16 mars 15:30-16:30 Tom Hutchcroft  (University of British Columbia)
The strange geometry of high-dimensional random forests

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Lieu : Salle 117-119

Résumé : The uniform spanning forest (USF) in the lattice Z^d, first studied by Pemantle (Ann. Prob. 1991), is defined as a limit of uniform spanning trees in growing finite boxes. Although the USF is a limit of trees, it might not be connected- Indeed, Pemantle proved that the USF in Z^d is connected if and only if d<5. Later, Benjamini, Kesten, Peres and Schramm (Ann. Math 2004) extended this result, and showed that the component structure of the USF undergoes a phase transition every 4 dimensions : For dimensions d between 5 and 8 there are infinitely many trees, but any two trees are adjacent ; for d between 9 and 12 this fails, but for every two trees in the USF there is an intermediary tree, adjacent to each of the them. This pattern continues, with the number of intermediary trees required increasing by 1 every 4 dimensions. In this talk, I will show that this is not the whole story, and for d>8 the USF geometry undergoes a qualitative change every time the dimension increases by 1.
Joint work with Yuval Peres. Based on http://arxiv.org/abs/1702.05780.

The strange geometry of high-dimensional random forests  Version PDF

Jeudi 16 mars 14:00-15:00 Zoltán Szabó  (École Polytechnique)
Minimax-optimal Distribution Regression

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Résumé : In my talk, I am going to focus on the distribution regression problem (DRP) : we regress from probability measures to Hilbert-space valued outputs, where the input distributions are only available through samples (this is the ’two-stage sampled’ setting). Several important statistical and machine learning tasks can be phrased within this framework including point estimation tasks without analytical solution (such as hyperparameter or entropy estimation) and multi-instance learning. However, due to the two-stage sampled nature of the problem, the theoretical analysis becomes quite challenging : to the best of our knowledge the only existing method with performance guarantees to solve the DRP task requires density estimation (which often performs poorly in practise) and the distributions to be defined on a compact Euclidean domain. We present a simple, analytically tractable alternative to solve the DRP task : we embed the distributions to a reproducing kernel Hilbert space and perform ridge regression from the embedded distributions to the outputs. Our main contribution is to prove that this scheme is consistent in the two-stage sampled setup under mild conditions : we present an exact computational-statistical efficiency tradeoff analysis showing that the studied estimator is able to match the one-stage sampled minimax-optimal rate. This result answers a 17-year-old open question, by establishing the consistency of the classical set kernel [Haussler, 1999 ; Gaertner et. al, 2002] in regression. We also cover consistency for more recent kernels on distributions, including those due to [Christmann and Steinwart, 2010]. The practical efficiency of the studied technique is demonstrated in supervised entropy learning and aerosol prediction using multispectral satellite images.

Minimax-optimal Distribution Regression  Version PDF

Jeudi 9 mars 14:00-15:00 Guillaume Poly  (Université de Rennes 1)
Universalité des longueurs de courbes nodales aléatoires

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Notes de dernières minutes : Dans cet exposé nous présenterons quelques modèles de fonctions multivariées aléatoires issus de la physique et aborderons la question du volume de leurs courbes de niveau. Nous expliquerons en quoi l’asymptotique de ces quantités est universelle et ne dépend pas des lois choisies pour générer l’aléa du modèle.

Universalité des longueurs de courbes nodales aléatoires  Version PDF

Jeudi 2 mars 14:00-15:00 Paul Doukhan  (Université de Cergy)
Chaluts à temps discret

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Lieu : Salle 117-119

Résumé : The talk essentially aims at describing the ongoing joint work with Silvia Lopes, Adam Jakubowski, and Donatas Surgailis. See file attached

Chaluts à temps discret  Version PDF

Jeudi 23 février 14:00-15:00 Matan Harel  (IHÉS)
Discontinuity of the phase transition for the planar random-cluster and Potts models with q>4

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Lieu : salle 117-119

Résumé : The ferromagnetic q-Potts Model is a classical spin system in which one of q colors is placed at every vertex of a graph and assigned an energy proportional to the number of monochromatic neighbors. It is highly related to the random-cluster model, which is a dependent percolation model where a configuration is weighted by q to the power of the number of clusters. Through non-rigorous means, Baxter showed that the phase transition is first-order whenever q>4 - i.e. there exists multiple Gibbs states at criticality. We provide a rigorous proof of the second claim. Like Baxter, our proof uses the correspondence between the above models and the six-vertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter’s formula for the correlation length of the models at criticality. This is joint work with Hugo Duminil-Copin, Maxemine Gangebin, Ioan Manolescu, and Vincent Tassion.

Discontinuity of the phase transition for the planar random-cluster and Potts models with q>4  Version PDF