## Mokameeting du 17 avril 2019 / Georgina HALL

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).

Venez nombreux!

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.

## Mokameeting du 13 mars 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:

Titre: Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces
Résumé:  In this talk, I will present a new ansatz space for the general symmetric multi-marginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from combinatorial in both N and  to (N+1), where  is the number of marginal states and N the number of marginals. These results were established in collaboration with Gero Friesecke.

Exposé de  Simone di Marino:

Titre: Duality in entropic optimal transport: a priori estimates and applications
Résumé: We want to explore a different approach to the duality in the entropic optimal transport, much more in the spirit of optimal transport, which is different from the usual techniques coming from the Schrodinger problem. This will result in consistent a priori estimates, which are consistent in the limit $\ep \to 0$. As a byproduct we prove that the IPFP algorithm is converging also in the multimarginal case.

## MOKAMEETING du 27 FEVRIER 10H-12H INRIA PARIS SALLE A315

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”

titre: Des liens et des interactions entre la géometrie convexe et la géometrie complexe
Résumé:

Je vais commencer avec une introduction aux variétés complexes:
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.

## Mokameeting du 20 février 2019: Mok@Dauphine!

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
16h10-16h30: pause
16h30-17h20: Laurent COHEN
17h20-18h10: Bruno BOUCHARD

Venez nombreux!

• 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.

## Mokameeting du 23 janvier 2019 / Pierre WEISS et Luca NENNA

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

Résumé:

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

Résumé:

This talk is devoted to variational problems on the set of probability
measures which involve optimal transport between unequal dimensional
spaces. In particular, we study the minimization of a functional consisting
of the sum of a term reflecting the cost of (unequal dimensional) optimal
transport between one fixed and one free marginal, and another functional
of the free marginal (of various forms). Motivating applications include
Cournot-Nash equilibria where the strategy space is lower dimensional
than the space of agent types. For a variety of different forms of the term
described above, we show that a nestedness condition, which is known to
yield much improved tractability of the optimal transport problem, holds
for any minimizer. Depending on the exact form of the functional, we
exploit this to find Monge-Ampère type equations characterising solutions,
prove convergence of an iterative scheme to compute the solution, and
prove regularity results. This a joint work with Brendan Pass.

## Mokameeting du 5 décembre 2018 / Julien RABIN et Mathurin MASSIAS

Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 5 décembre à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle Jacques-Louis Lions 1 (attention, salle inhabituelle!).

Nous aurons le plaisir d’écouter Julien RABIN (Université de Caen)  et Mathurin MASSIAS (Télécom PARISTECH / INRIA PARIETAL).

• Exposé de Julien RABIN:

Titre: Semi-Discrete Optimal Transport in Patch Space for Enriching Gaussian Textures

Résumé: A bilevel texture model is proposed, based on a local transform of a Gaussian random field. The core of this method relies on the optimal transport of a continuous Gaussian distribution towards the discrete exemplar patch distribution. The synthesis then simply consists in a fast post-processing of a Gaussian texture sample, boiling down to an improved nearestneighbor patch matching, while offering theoretical guarantees on statistical compliancy.

• Exposé de Mathurin MASSIAS

Titre: Dual extrapolation for sparse Generalized Linear models

Résumé: Generalized Linear Models (GLM) are a wide class of regression and classification models, where the predicted variable is obtained from a linear combination of the input variables.
For statistical inference in high dimensions, sparsity inducing regularization have proven useful while offering statistical guarantees.
However, solving the resulting optimization problems can be challenging: even for popular iterative algorithms such as coordinate descent, one needs to loop over a large number of variables.
To mitigate this, techniques known as screening rules and working sets diminish the size of the optimization problem at hand, either by progressively removing variables, or by solving a growing sequence of smaller problems.
For both of these techniques, significant variables are identified by convex duality.
In this talk, we show that the dual iterates of a GLM exhibit a Vector AutoRegressive (VAR) behavior after support identification, when the primal problem is solved with proximal gradient or cyclic coordinate descent.
Exploiting this regularity one can construct dual points that offer tighter control of optimality, enhancing the performance of screening rules and helping to design a competitive working set algorithm.

## Mokameeting du 7 novembre 2018 / Xiaolu TAN et Claire LAUNAY

Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 7 novembre à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle A415.

Nous aurons le plaisir d’écouter Xiaolu TAN (Université Paris-Dauphine) et Claire LAUNAY (Université Paris-Descartes).

• Exposé de Xiaolu TAN
Title: On the optimal planning problem for a class of Mean Field Games
Abstract: In the context of a generalized version of mean field games, with possible control of the diffusion coefficient, we consider the planning problem introduced by P.L. Lions: given a pair $(\mu,\nu)$ of starting and target probability measures on the state
space, find a specification of the game problem which induces $\nu$ at the mean field game equilibrium. We provide conditions on $(\mu,\nu)$ which guarantee existence of solutions. We next introduce an optimal planification problem, which is then reduced to a stochastic control problem of the McKean-Vlasov SDEs.

• Exposé de Claire LAUNAY

Titre: Processus ponctuels déterminantaux et images : quelques exemples d’application

Résumé: Les processus ponctuels déterminantaux (PPD) permettent de modéliser le caractère répulsif de certains ensembles de points. Ils capturent les corrélations négatives : plus deux points sont similaires, moins il est probable qu’ils soient échantillonnés simultanément. Ces processus ont donc tendance à générer des ensembles de points diversifiés ou éloignés les uns des autres. Contrairement à d’autres processus répulsifs, ceux-ci ont l’avantage d’être entièrement déterminés par leur noyau, leurs moments sont tous connus et il existe un algorithme exact pour les échantillonner. Lors de cet exposé, je présenterai les processus ponctuels déterminantaux dans un cadre discret général puis dans celui des images : un cadre 2D, stationnaire et périodique. Nous avons étudié les propriétés de répulsion de tels processus, notamment en utilisant les modèles shot noise, propriétés qui s’avèrent intéressantes pour synthétiser des micro-textures. Je présenterai également comment les processus déterminantaux peuvent s’appliquer au sous-échantillonnage d’une image dans l’espace des patchs.

## Mokameeting du 3 octobre 2018 / Paul Pegon: Introduction au transport branché

Le prochain séminaire de l’équipe Mokaplan aura lieu le mercredi 3 octobre à 10h30 à l’INRIA Paris (2 rue Simone Iff) en salle A415.

Nous aurons le plaisir d’écouter Paul Pegon (Université Paris-Dauphine / MOKAPLAN):

Titre: Introduction au transport branché

Résumé: Dans cet exposé, j’introduirai le problème du transport branché en en présentant les deux principaux modèles (eulérien et lagrangien). J’expliquerai quelques propriétés qualitatives des solutions, ainsi qu’une manière de calculer des solutions approchées. Enfin, je ferai le point de la recherche sur les structures fractales qui apparaissent en transport branché.

## MOKAMEETING September 19th 2018, 10h-12h, Room A415

Room A415, 10:00 – 12:00

• Joseph Budin, Wasserstein Barycenters for Persistence Diagrams
• Jean Alaux, Optimal Transport Geometry and Text Analysis
• Thibault Séjourné, Sinkhorn Entropies and Divergences

## MOKAMEETING 22 Aout 2018

• Lucas Martinet (INRIA) – Sinkhorn and parallelism
• Vincent Duval (INRIA) – tightness of Lasserre relaxations on the multi-dimensional torus.