In this talk, I will present recent work where such blessings of dimensionality are exploited. In particular, I will show (1) the exact characterization of a widely-used spectral method for nonconvex statistical estimation; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze nonconvex optimization algorithms for high-dimensional inference and learning. In these problems, asymptotic methods not only clarify some of the fascinating phenomena that emerge with high-dimensional data, they also lead to optimal designs that significantly outperform heuristic choices commonly used in practice.
Oct 02 2019
Sep 30 2019
Une preuve du théorème de Caffarelli via la régularisation entropique
Le théorème de contraction de Caffarelli (2001) énonce que le transport optimal de la mesure Gaussienne sur une mesure uniformément log-concave est globalement lipschitz. Dans cet exposé, je présenterai une nouvelle preuve, basée sur la régularisation entropique du transport optimal et une caractérisation variationnelle des transport lipschitz, dûe à Gozlan et Juillet. Travail en collaboration avec Nathael Gozlan et Maxime Prod’homme.
Sep 30 2019
Jul 17 2019
MOKAMEETING 25 SEPTEMBER 14H . ROOM JLL INRIA PARIS
HongKai Zhao, UC Irvine.
Jun 12 2019
Abstract: Optimal transport (OT) is a method of measuring distances between probability distributions that has found numerous applications in machine learning, computer graphics, image processing, and others. However, research has focused largely on computing transport distances quickly in great generality, and less on developing better algorithms for problems with inherent structure.
In this talk, I hope to provide perspective on how additional knowledge and constraints on the distributions or the underlying domain help in designing better algorithms. I will discuss two directions: (1) learning problems that have natural OT interpretations and lead to simple constraints and efficient algorithms, and (2) connections between the semi-discrete transport problem and classical learning problems.
Apr 01 2019
Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 17 avril à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle Jacques-Louis Lions 1.
Nous aurons le plaisir d’écouter Georgina HALL (INSEAD).
Title: Sum of squares optimization: fundamentals, applications, and recent scalability developments
Abstract: The problem of optimizing over nonnegative polynomials, and its dual formulation – optimizing over the set of moments that have a representing measure – are optimization problems that naturally arise in a variety of applications. In the first part of this talk, we will review a number of these applications in control, statistics, and probability, among others. We will also discuss how these problems can be tackled using sum of squares optimization, a subclass of optimization problems whose computational backbone is semidefinite programming. In the second part of this talk, we will focus on a major challenge that has limited the dissemination of sum of squares optimization within more applied fields: scalability. We will briefly review a few methods that have been developed to curb this issue, focusing on methods that replace the underlying semidefinite program with cheaper conic programs.
Mar 03 2019
Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 13 mars à 10h30 à l’université Paris-Dauphine (Place du Maréchal de Lattre de Tassigny), en salle B207.
Nous aurons le plaisir d’écouter Daniela Vögler (TUM, Munich) et Simone di Marino (Indam, SNS, Pisa).
Exposé de Daniela Vögler:
Exposé de Simone di Marino:
Feb 11 2019
MOKAMEETING du 27 FEVRIER 10H-12H INRIA PARIS SALLE A315
“Nous accueillerons le Prof. Yanir Rubinstein (U. Maryland) qui nous fera une introduction :
“Interactions between convex geometry and complex geometry”
définitions, thèmes de recherches courants, conjectures et histoire.
Puis, je vais décrire quelques relations intéressantes entre variétés complexes et géométrie convexe, transport optimal, EDP de type
Monge-Ampère, et inégalités fonctionnelles.
Feb 03 2019
Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 20 février à 14h30 à l’université Paris-Dauphine (Place du Maréchal de Lattre de Tassigny), en salle P509.
Pour cette édition spéciale, nous aurons le plaisir d’écouter 4 exposés au cours de l’après-midi. Le programme est le suivant:
14h30-15h20: Marc HOFFMANN
15h20-16h10: Olga MULA
16h30-17h20: Laurent COHEN
17h20-18h10: Bruno BOUCHARD
- Exposé de Marc HOFFMANN:
Titre: Estimation statistique pour des modèles structurés en âge dans une limite grande population
Résumé: Motivated by improving mortality tables from human demography databases, we investigate statistical inference of a stochastic age-evolving density of a population alimented by time inhomogeneous mortality and fertility. Asymptotics are taken as the size of the population grows within a limited time horizon: the observation gets closer to the solution of the Von Foerster Mc Kendrick equation, and the difficulty lies in controlling simultaneously the stochastic approximation to the limiting PDE in a suitable sense together with an appropriate parametrisation of the anisotropic solution.
In this setting, we prove new concentration inequalities that enable us to implement the Goldenshluger-Lepski algorithm and derive oracle inequalities. We obtain minimax optimality and adaptation over a wide range of anisotropic Hölder smoothness classes.
- Exposé d’Olga MULA:
Titre: Nonlinear reduced models and state estimation
Abstract: In this talk, we present an overview and some recent results on the problem of reconstructing in real time the state of a physical system from available measurement observations and the knowledge of a physical PDE model. Contrary to classical inverse problem approaches where one seeks for the parameters of the PDE that best satisfy the measurements, we use the PDE models to learn fast reconstruction mappings which satisfy certain optimality properties. The high dimensionality of the problems that arise combined with the very different nature of the potential applications (air pollution, hemodynamics, nuclear safety to name a few) demand the development of compression, optimisation and learning strategies based on sound mathematical grounds. In the talk, we will present recent results on optimal affine algorithms and highlight the prominent role of reduced order modelling of PDEs. However, in its classical formulation, reduced order modelling involves the construction of linear spaces which makes it not suitable to treat hyperbolic problems. We will outline recent results on an approach involving nonlinear mappings to mitigate this obstruction.
- Exposé de Laurent COHEN:
Titre: Méthodes géodésiques pour la segmentation d’images
Résumé: Tubular and tree structures appear very commonly in biomedical images like vessels, microtubules or neuron cells. Minimal paths have been used for long as an interactive tool to segment these structures as cost minimizing curves. The user usually provides start and end points on the image and gets the minimal path as output. These minimal paths correspond to minimal geodesics according to some adapted metric. They are a way to find a (set of) curve(s) globally minimizing the geodesic active contours energy. Finding a geodesic distance can be solved by the Eikonal equation using the fast and efficient Fast Marching method. Introduced first as a way to find the global minimum of a simplified active contour energy, we have recently extended these methods to cover all kinds of active contour energy terms. For example a new way to penalize the curvature in the framework of geodesic minimal paths was introduced, leading to more natural results in vessel extraction for example.
- Exposé de Bruno BOUCHARD:
Titre: Quenched mass transport of particles towards a target
Résumé: We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost-surely transported towards a given target, along the paths of a stochastic differential equation.
Jan 14 2019
Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 23 janvier à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle Jacques-Louis Lions 1.
Nous aurons le plaisir d’écouter Pierre WEISS (CNRS/ ITAV) et Luca NENNA (Université Paris-Sud).
- Exposé de Pierre WEISS:
Titre: A convergence analysis of some exchange algorithms
Exchange algorithms are a class of optimization methods to solve semi-infinite programs: optimization problems over a finite dimensional variable with an infinite number of constraints. These problems arise naturally in a variety of situations. We will focus here on their use for the resolution of inverse problems without discretization of the underlying domain. A typical example is super-resolution by using a total variation regularizer. Our main result states the linear convergence rate of the method under technical assumptions such as a non degeneracy condition.
- Exposé de Luca NENNA:
Titre: Unequal Dimensional Optimal Transport, Monge-Ampère equations and beyond