## Prochainement

Jeudi 14 décembre 14:00-15:00 Bertrand Thirion (INRIA)
Statistical Testing for high-dimensional Models : Leveraging data structure for higher efficiency and accuracy

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iCal

Lieu : salle 117/119 du bâtiment 425

Résumé : In many scientific applications, increasingly-large datasets are being acquired to describe more accurately biological or physical phenomena. While the dimensionality of the resulting measures has increased, the number of samples available is often limited, due to physical or financial limits. This results in impressive amounts of complex data observed in small batches of samples. A question that arises is then : what features in the data are really informative about some outcome of interest ? This amounts to inferring the relationships between these variables and the outcome, conditionally to all other variables. Providing statistical guarantees on these associations is needed in many fields of data science, where competing models require rigorous statistical assessment. Yet reaching such guarantees is very hard.
In this presentation, we will first motivate the quest for inference models by examples from applied statistical problems. We will them review existing solutions, together with their strengths and weaknesses and outline promising directions. We will eventually discuss how to introduce structure in such models while retaining statistical guarantees.

Statistical Testing for high-dimensional Models : Leveraging data structure for higher efficiency and accuracy  Version PDF

## Passés

Jeudi 7 décembre 14:00-15:00 Benoît Collins (Université de Kyoto)
Liberté asymptotique forte pour des permutations aléatoires

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Lieu : salle 117/119 du bâtiment 425

Résumé : n by n permutation matrices act naturally on the (n − 1)-dimensional vector subspace of C^n of vectors whose components add up to zero. We prove that random independent permutations, viewed as operators on this vector subspace, are asymptotically strongly free with high probability. While this is a counterpart of a previous result by the presenter and Male in the case of a uniform distribution on unitary matrices, the techniques required for random permutations are very different, and rely on the development of a matrix version of the theory of non-backtracking operators. This is joint work with Charles Bordenave.

Liberté asymptotique forte pour des permutations aléatoires  Version PDF
Jeudi 30 novembre 14:00-15:00 Sigurður Stefánsson (University of Iceland)
Random outerplanar maps and stable looptrees

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Lieu : salle 117/119 du bâtiment 425

Résumé : An outerplanar map is a drawing of a planar graph in the sphere which has the property that there is a face in the map such that all the vertices lie on the boundary of that face. A random outerplanar map is defined by assigning non-negative weights to each face of a map. Caraceni showed that uniform outerplanar maps (all weights equal to 1) with an appropriately rescaled graph distance converge to Aldous’ Brownian tree in the Gromov-Hausdorff sense. This result was generalized by Stufler who showed that the same holds under some moment conditions. I show, in joint work with Stufler, that for certain choices of weights the maps converge towards the alpha-stable looptree, which was recently introduced by Curien and Kortchemski. Our approach relies on the fact that outerplanar maps may be viewed as trees in which each vertex is a dissection of a polygon. Dissections of polygons are further in bijection with trees which allows us to relate the random outerplanar maps to the model of simply generated trees which is understood in detail.

Random outerplanar maps and stable looptrees  Version PDF
Jeudi 23 novembre 14:00-15:00 Randal Douc (Télécom SudParis)
Posterior consistency for partially observed Markov models

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Lieu : Salle 117/119 du bâtiment 425

Résumé : We establish the posterior consistency for a parametrized family of partially observed, fully dominated Markov models. The prior is assumed to assign positive probability to all neighborhoods of the true parameter, for a distance induced by the expected Kullback-Leibler divergence between the family members’ Markov transition densities. This assumption is easily checked in general. In addition, we show that the posterior consistency is implied by the consistency of the maximum likelihood estimator. The result is extended to possibly non-compact parameter spaces and non-stationary observations. Finally, we check our assumptions on a linear Gaussian model and a well-known stochastic volatility model.
Joint work with Francois Roueff and Jimmy Olsson.

Posterior consistency for partially observed Markov models  Version PDF
Jeudi 16 novembre 15:30-16:30 Christophe Texier (Université Paris-Sud)
Truncated linear statistics associated with the eigenvalues of random matrices

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Lieu : salle 117/119 du bâtiment 425

Notes de dernières minutes : Given a certain invariant random matrix ensemble characterised by the joint probability distribution of eigenvalues $P(\lambda_1,\cdots,\lambda_N)$, the study of linear statistics of the eigenvalues $L=\sum_{i=1}^N f(\lambda_i)$, where $f(\lambda)$ is a known function, has played an important role in many applications of random matrix theory. I will discuss the distribution of truncated linear statistics of the form $\tilde{L}=\sum_{i=1}^{N_1} f(\lambda_i)$, when the sum runs over a fraction of the eigenvalues ($N_1<N$). By using the Coulomb gas technique, the large deviation function controlling the distribution of such sums in the limit of large $N$, with $0 < N_1/N < 1$ fixed, will be analysed. Two situations will be considered leading to two different universal scenarii : -# the case where the truncated linear statistics is restricted to the largest (or smallest) eigenvalues. We have shown that the constraint that $\tilde{L}=\sum_{i=1}^{N_1} f(\lambda_i)$ is fixed drives an infinite order phase transition in the underlying Coulomb gas. This transition corresponds to a change in the density of the gas, from a density defined on two disjoint intervals to a single interval. In this latter case the density presents a logarithmic divergence inside the bulk. -# the second situation is the case without further restriction on the ordering of the eigenvalues contributing to the truncated linear statistics (this can be viewed as a new ensemble which is related, but not equivalent, to the thinned ensembles’’ introduced by Bohigas and Pato). In this case, a region opens in the phase diagram of the Coulomb gas, where the large deviation function is mostly controlled by entropy (in particular this induces a change in the scaling of the relative fluctuations of the truncated linear statistics, from the usual $1/N$ for $N_1=N$, to $1/\sqrt{N}$ when $N_1<N$). Our analysis relies on the mapping on a problem of $N_1$ fictitious non-interacting fermions in $N$ energy levels, which can exhibit both positive and negative effective (absolute) temperatures.

Truncated linear statistics associated with the eigenvalues of random matrices  Version PDF
Jeudi 16 novembre 14:00-15:00 Grégory Schehr (Université Paris-Sud)
Noninteracting trapped fermions : from random matrices to the Kardar-Parisi-Zhang equation

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Lieu : salle 117/119 du bâtiment 425

Résumé : I will consider a system of N one-dimensional free fermions confined by a harmonic well. At zero temperature (T=0), this system is intimately connected to random matrices belonging to the Gaussian Unitary Ensemble. In particular, the density of fermions has, for large N, a finite support and it is given by the Wigner semi-circular law. Besides, close to the edges of the support, the quantum fluctuations are described by the so-called Airy-Kernel (which plays an important role in random matrix theory). What happens at finite temperature T ? I will show that at finite but low temperature, the fluctuations close to the edge, are described by a generalization of the Airy kernel, which depends continuously on temperature. Remarkably, exactly the same kernel arises in the exact solution of the Kardar-Parisi-Zhang (KPZ) equation in 1+1 dimensions at finite time. I will also discuss extensions of these results to fermions in higher dimensions.

Noninteracting trapped fermions : from random matrices to the Kardar-Parisi-Zhang equation  Version PDF
Jeudi 9 novembre 14:00-15:00 Cristina Butucea (Université Paris-Est Marne-la-Vallée)
Tests non paramétriques de grandes matrices de covariance

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Lieu : salle 117/119 du bâtiment 425

Résumé : Dans un modèle de n vecteurs indépendants gaussiens de dimension p, on s’intéresse à la détection des corrélations significatives. Nous supposons que la matrice de covariance appartient à un certain ellipsoïde et proposons un test basé sur une U-statistique d’ordre 2 pondérée de manière optimale. Nous donnons les vitesses de séparation minimax ainsi que les constantes exactes asymptotiques (en n et p). Des vitesses plus rapides sont obtenues pour les matrices de Toeplitz. Dans ce dernier cas nous discutons des résultats non asymptotiques.

Tests non paramétriques de grandes matrices de covariance  Version PDF
 Département de Mathématiques Bâtiment 425 Faculté des Sciences d'Orsay Université Paris-Sud F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56