CERMICS Colloquium

Forthcoming sessions

Laure Saint-Raymond (ENS de Lyon)

Monday, December 18th, 14h – Salle de séminaire du CERMICS

Forçage quasi-résonnant d’ondes internes dans un domaine avec topographie
(en collaboration avec Yves Colin de Verdière)

La stratification en densité dans un fluide incompressible est responsable de la propagation d’ondes internes. Dans les domaines avec topographie, ces ondes présentent des propriétés intéressantes. En particulier, les expériences numériques et de laboratoire montrent qu’en 2D, pour des fréquences génériques de forçage, ces ondes se concentrent sur des attracteurs (voir Dauxois et al). Au niveau mathématique, ce comportement peut être analysé avec des outils de théorie spectrale et d’analyse microlocale.

 

Yann Brenier (Ecole Polytechnique)

Monday, March 5th, 10h30 – Salle de séminaire du CERMICS

Le problème de Cauchy pour les systèmes de lois de conservation entropiques traité par minimisation convexe

On montre que, pour les systèmes du premier ordre de lois de conservation avec une entropie strictement convexe, en particulier pour la très simple équation de Burgers sans viscosité, il est possible d’attaquer le problème de Cauchy par un problème de minimisation convexe approprié, assez semblable a un problème de transport optimal ou de jeu à champ moyen. Dans le cas général, on montre que les solutions lisses, sans choc, peuvent être retrouvées sur des intervalles de temps assez petits. Dans la situation particulière de l’équation de Burgers, nous montrons de plus que toute “solution entropique” (au sens de Kruzhkov), y compris avec choc, peut être reconstituée, sur des intervalles de temps arbitrairement longs.

 

Past sessions

Daniel Kuhn (EPFL)

Wednesday, October 4th, 15h30 – Salle de séminaire du CERMICS

Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations

We consider stochastic programs where the distribution of the uncertain parameters is only observable through a finite training dataset. Using the Wasserstein metric, we construct a ball in the space of (multivariate and non-discrete) probability distributions centered at the uniform distribution on the training samples, and we seek decisions that perform best in view of the worst-case distribution within this Wasserstein ball. The state-of-the-art methods for solving the resulting distributionally robust optimization problems rely on global optimization techniques, which quickly become computationally excruciating. In this paper we demonstrate that, under mild assumptions, the distributionally robust optimization problems over Wasserstein balls can in fact be reformulated as finite convex programs – in many interesting cases even as tractable linear programs. Leveraging recent measure concentration results, we also show that their solutions enjoy powerful finite-sample performance guarantees. Our theoretical results are exemplified in mean-risk portfolio optimization as well as uncertainty quantification.

 

Arnak Dalalyan (ENSAE)

Thursday, June 15th 2017, 10h

On the Exponentially Weighted Aggregate with the Laplace Prior

We study the statistical behaviour of the Exponentially Weighted Aggregate (EWA) in the problem of high-dimensional regression with fixed design. Under the assumption that the underlying regression vector is sparse, it is reasonable to use the Laplace distribution as a prior. The resulting estimator and, specifically, a particular instance of it referred to as the Bayesian lasso, was already used in the statistical literature because of its computational convenience, even though no thorough mathematical analysis of its statistical properties was carried out. The present work fills this gap by establishing sharp oracle inequalities for the EWA with the Laplace prior. These inequalities show that if the temperature parameter is small, the EWA with the Laplace prior satisfies the same type of oracle inequality as the lasso estimator does, as long as the quality of estimation is measured by the prediction loss. Extensions of the proposed methodology to the problem of prediction with low-rank matrices are considered.
Reference: Arnak S. Dalalyan, Edwin Grappin, Quentin Paris, arXiv:1611.08483.

Mitchell Luskin (University of Minnesota)

Monday October 17th, 2016, 14h

Mathematical Modelling of Incommensurate Materials

Incommensurate materials are found in crystals, liquid crystals, and quasi-crystals. Stacking a few layers of 2D materials such as graphene and molybdenum disulfide, for example, opens the possibility to tune the elastic, electronic, and optical properties of these materials. One of the main issues encountered in the mathematical modeling of layered 2D materials is that lattice mismatch and rotations between the layers destroys the periodic character of the system. This leads to complex commensurate-incommensurate transitions and pattern formation. Even basic concepts like the Cauchy-Born strain energy density, the electronic density of states, and the Kubo-Greenwood formulas for transport properties have not been given a rigorous analysis in the incommensurate setting. New approximate approaches will be discussed and the validity and efficiency of these approximations will be examined from mathematical and numerical analysis perspectives.

Common seminar CERMICS/IMAGINE

Tuesday June 7th, 10h

10h : Mathieu Aubry (IMAGINE, École des Ponts ParisTech), Representing 3D models for matching and retrieving (abstract, slides),
11h : Francis Bach (Inria et ENS), Beyond stochastic gradient descent for large-scale machine learning (abstract, slides).

Emilie Kaufmann (CNRS)

Wednesday March 30th, 2016, 14h

Multi-armed bandit models: a tutorial (slides)

Reinforcement learning refers to the situation in which an agent learns to optimaly interact with his environment by trying to maximize rewards received during the interaction. In the simplest case, the interaction consists in repeatedly choosing actions that lead to some (possibly random) payoff. This situation can be described by a multi-armed bandit model. Even if this name refers to a gambler who tries to maximize its gain by sequentially choosing the arm of which slot-machine (or one-armed bandit) he wants to draw, bandit models were introduced to model clinical trials, and have been recently successfully applied to online content optimization. In this tutorial, I will present different types of bandit models and related algorithms (for choosing which action to play based on past observation. Under each of these model assumptions, I will try to define some optimality criterion, and introduce the tools to build optimal algorithms.

Hugo Touchette (National Institute for Theoretical Physics)

Monday November 30th, 2015, 14h

Théorie des grandes déviations : des mathématiques à la physique. (slides)

Cet exposé se veut un survol général de la théorie des grandes déviations et de ses applications en physique statistique. La première partie rappellera quelques sources intéressantes de cette théorie issues des mathématiques (Sanov, Varadhan) et de la physique (Boltzmann, Einstein, Lanford), et exposera les résultats essentiels de cette théorie. Dans la deuxième partie, quelques applications physiques seront présentées afin de démontrer l’utilité de la théorie des grandes déviations en physique statistique, tant pour étudier les systèmes à l’équilibre que hors équilibre. L’exposé ne supposera aucune connaissance préalable de physique statistique.