# Séminaire et Groupe de travail PAS

Organisateurs : Erwan Saint-Loubert Bié et Christoph Kriegler
Les exposés ont lieu le mardi à 14h45 en salle 2222 du bâtiment de mathématiques (consulter le plan d'accès au laboratoire).

### Mars 2019

• Mardi 05 mars 2019 - Jean BARBIER, ENS Paris

Phase transitions in high-dimensional estimation and learning

I will present a unified and mathematically rigorous framework allowing to locate the information-theoretic and algorithmic limits in high-dimensional generalized linear models (GLMs). This allows to draw « phase diagrams » as in physics, the phases being associated to different algorithmic behaviors. The GLM includes as special cases plethora of important models in signal processing (compressed-sensing, phase retrieval etc), communications, but also in learning such as the famous perceptron neural network. Many special cases of GLMs have been analyzed in the statistical physics literature, in particular thanks to the heuristic replica method developed in the context of spin glasses. I will discuss recent mathematical tools that vindicate the statistical approach, as well as recent findings about the rich algorithmic behaviors encountered in such models.

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### Février 2019

• Mardi 19 février 2019 - Adrián Gonzalez-Perez (KU Louvain, Belgique)

Lp-boundedness of multipliers on von Neumann algebras

We will present some results for completely bounded Fourier multipliers on the Lp space of a group von Neumann algebra. Such results are a generalization the Marcinkiewicz theorem and the spectral Hörmander-Mikhlin theorems. These theorems give bounded Lp-multipliers when the symbol has a finite number of suitably bounded, derivatives. As an application we will see that for conditionally negative functions arising from 1-cocycles we have a Hörmander-Mikhlin multiplier result with the smoothness order controlled by the rank of the cocycle. A crucial novelty is that we use a principle of boundedness of Fourier multipliers by noncommutative maximal operators, which gives extra results beyond the spectral case.

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• Mardi 12 février 2019 - Ethan ANDERES, UC Davis and IHES

Gravitational wave and lensing inference from the CMB polarization

In the last decade cosmologists have spent a considerable amount of effort mapping the radially-projected large-scale mass distribution in the universe by measuring the distortion it imprints on the Cosmic Microwave Background (CMB). Indeed, all the major surveys of the CMB produce estimated maps of the projected gravitational potential generated by mass density fluctuations over the sky. These maps contain a wealth of cosmological information and, as such, are an important data product of CMB experiments. However, the most profound impact from CMB lensing studies may not come from measuring the lensing effect, per se, but rather from our ability to remove it, a process called delensing''. This is due to the fact that lensing, along with emission of millimeter wavelength radiation from the interstellar medium in our own galaxy, are the two dominant sources of foreground contaminants for primordial gravitational wave signals in the CMB polarization. As such delensing, i.e. the process of removing the lensing contaminants, and our ability to either model or remove galactic foreground emission sets the noise floor on upcoming gravitational wave science. In this talk we will present a complete Bayesian solution for simultaneous inference of lensing, delensing and gravitational wave signals in the CMB polarization as characterized by the tensor-to-scalar ratio r parameter. Our solution relies crucially on a physically motivated re-parameterization of the CMB polarization which is designed specifically, along with the design of the Gibbs Markov chain itself, to result in an efficient Gibbs sampler---in terms of mixing time and the computational cost of each step---of the Bayesian posterior. This re-parameterization also takes advantage of a newly developed lensing algorithm, which we term LenseFlow, that lenses a map by solving a system of ordinary differential equations. This description has conceptual advantages, such as allowing us to give a simple non-perturbative proof that the lensing determinant is equal to unity in the weak-lensing regime. The algorithm itself maintains this property even on pixelized maps, which is crucial for our purposes and unique to LenseFlow as compared to other lensing algorithms we have tested. It also has other useful properties such as that it can be trivially inverted (i.e. delensing) for the same computational cost as the forward operation, and can be used for fast and exact likelihood gradients with respect to the lensing potential. Incidentally, the ODEs for calculating these derivatives are exactly analogous to the backpropagation techniques used in deep neural networks but are derived in this case completely from ODE theory.

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### Janvier 2019

• Mardi 22 janvier 2019 - Léonard Cadilhac (Université de Caen)

Inégalités de Khintchine non-commutatives dans les espaces symétriques

Les inégalités de Khintchine non-commutatives ont été formulées et démontrées pour la première fois par Lust-Piquard en 1981. Ces inégalités sont fréquemment utilisées en analyse harmonique non-commutative et ont été généralisées dans deux directions : pour savoir dans quels espaces (Lorentz, Orlicz) et pour quelles suites de variables (unitaires de Haar libres) elles restent vraies. Dans cet exposé, je commencerai par brièvement rappeler les inégalités de Khintchine classiques puis donner un aperçu des notions de bases de l’intégration non-commutative. Je terminerai par énoncer et donner des idées de preuve de nouveaux résultats sur les inégalités de Khintchine dans les interpolés d’espaces Lp.

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• Mardi 15 janvier 2019 - Abdallah Maichine (Marrakech, Maroc)

Lp-theory for Schrödinger operators with matrix potential

Consider a Schr\"odinger type operator with matrix potential of the form $\mathcal{A}u=div(Q\nabla u)-Vu$ acting on vector--valued functions $u=(u_1,\dots,u_m):\R^d\to\R^m$, where $Q$ is a bounded matrix map satisfying the ellipticity condition and $V$ a semi--definite positive matrix map. We associate a $C_0$-semigroup to realizations $A_p$ of $\mathcal{A}$ in $L^p(\R^d,\R^m)$, $1 Afficher le contenu... ### Octobre 2018 • Mercredi 10 octobre 2018 - Manon Costa, Institut Mathématique de Toulouse Renouvellement pour les processus de Hawkes Les processus de Hawkes modélisent les occurrences successives d'évènements auto-excitatif comme les répliques d'un séismes ou les décharges neuronales. Ils sont paramétrés par une fonction de reproduction$h$qui décrit l'impact d'un évènement initial sur la probabilité d'arrivée d'évènements futurs. Ils peuvent aussi inclure un phénomène d'inhibition et dans ce cas, la fonction de reproduction$h\$ prend des valeurs négatives. Le but de cet exposé est d'introduire les outils nécessaire à l'obtention d'inégalités de concentration pour cette classe de processus de Hawkes. Ce travail est en collaboration avec Carl Graham (Ecole Polytechnique), Laurence Marsalle (Lille), et Viet Chi Tran (Lille).

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