5 décembre 2018

Raphaël Tinarrage 
Homologie persistante

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Lieu : Bâtiment 307, salle 3L8

Résumé : Soit X un sous-ensemble fini d’un espace euclidien, donné par le résultat d’une expérience scientifique. Si l’on croit que X cache une structure topologique intéressante (par exemple s’il est proche d’une sous-variété M) et que l’on essaye de la comprendre, alors on dit que l’on fait de l’Analyse Topologique des Données. Plutôt que de reconstruire (au type d’homotopie près) la sous-variété sous-jacente M à partir de X (procédure instable et difficile en grande dimension), la théorie de l’homologie persistante permet d’estimer l’homologie (singulière) de M à partir de X, à travers ce que l’on appelle le diagramme de persistance de X. J’expliquerai dans cet exposé le formalisme algébrique dans lequel s’exprime cette théorie, avec des exemples de nature topologique.
Persistent homology
Let X be a finite subset of a Euclidean space, resulting from a scientific experiment. If one thinks that X hides an interesting topological structure (e.g. X is close to some submanifold M) and tries to understand it, then we say that one is doing Topological Data Analysis. Instead of reconstructing the homotopy type of the underlying submanifold M from X (unstable procedure and difficult in high dimension), the theory of persistent homology gives a way to estimate the (singular) homology of M from X, through what is called the persistence diagram of X. In this talk I will develop the algebraic setting of this theory, with topological examples.

Homologie persistante  Version PDF

Joseph Salmon (Université de Montpellier)
Generalized Concomitant Multi-Task Lasso for sparse multimodal regression

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Résumé : In high dimension, it is customary to consider Lasso-type estimators to enforce sparsity.
For standard Lasso theory to hold, the regularization parameter should be proportional to the noise level, which is often unknown in practice.
A remedy is to consider estimators such as the Concomitant Lasso, which jointly optimize over the regression coefficients and the noise level.
However, when data from different sources are pooled to increase sample size, noise levels differ and new dedicated estimators are needed.
We provide new statistical and computational solutions to perform het-eroscedastic regression, with an emphasis on brain imaging with magneto-and electroen-cephalography (M/EEG). When instantiated to de-correlated noise, our framework leads to an efficient algorithm whose computational cost is not higher than for the Lasso, but addresses more complex noise structures. Experiments demonstrate improved prediction and support identification with correct estimation of noise levels.
This is joint work with M. Massias, O. Fercoq and A. Gramfort.
Arxiv <https://arxiv.org/abs/1705.09778>
Python code <https://github.com/mathurinm/SHCL>

Generalized Concomitant Multi-Task Lasso for sparse multimodal regression  Version PDF

Vincent Vargas  (ENS Ulm)
An introduction to Liouville conformal field theory

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Lieu : Salle 3L8

Résumé : Liouville conformal field theory (LCFT hereafter), introduced by
Polyakov in his 1981 seminal work « Quantum geometry of bosonic strings »,
can be seen as a random version of the theory of Riemann surfaces. LCFT
in Polyakov’s work as a 2d version of the Feynman path integral with an
exponential interaction term. Since then, LCFT has emerged in a wide
variety of contexts in the physics literature and in particular recently
in relation with 4d supersymmetric gauge theories (via the AGT conjecture).
The purpose of this talk is to present in detail a rigorous probabilistic
construction of Polyakov’s path integral formulation of LCFT : the
construction is based on the Gaussian Free Field. If time permits, I will
also discuss the (conjectured) relation between the scaling limit of large
random planar maps and LCFT.

An introduction to Liouville conformal field theory  Version PDF