Séminaire de mathématiques appliquées (archives)

Abstract

A method for constructing signatures of random reparametrizations of periodic functions is presented.

The proposed signatures are functions, which contain information about the height and order of local extrema of the observation. In contrast to other statistical methods for reparametrized curves, the observations can be of different lengths and the construction does not involve aligning them.

The signature is shown to be stable with respect to changes in the distribution of reparametrizations and to enjoy standard CLT properties, including in the case of dependent observations.

The positioning of a vehicle based on magnetic signals is the industrial application which motivated this work.

Heterogeneous problems that take place at multiple scales are ubiquitous in science and engineering. Examples are wind turbines made from composites or groundwater flow relevant e.g., for the design of flood prevention measures. However, finite element or finite volume methods require an often prohibitively large amount of computational time for such tasks. Multiscale methods that are based on ansatz functions which incorporate the local behavior of the (numerical) solution of the partial differential equations (PDEs) have been developed to tackle these heterogeneous problems.

Dependence measures based on reproducing kernel Hilbert spaces, also known as Hilbert-Schmidt Independence Criterion and denoted HSIC, are widely used to statistically decide whether or not two random vectors are dependent. Recently, non-parametric HSIC-based statistical tests of independence have been performed. However, these tests lead to the question of the choice of the kernels associated to the HSIC. In particular, there is as yet no method to objectively select specific kernels with theoretical guarantees in terms of first and second kind errors. One of the main contributions of this work is to develop a new HSIC-based aggregated procedure which avoids such a kernel choice, and to provide theoretical guarantees for this procedure.

In this talk, I will consider a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system, which arises in population dynamics. This system has entropy dissipation properties on which one can rely to design a robust and convergent numerical scheme for its numerical simulation. In terms of numerical analysis, I will present discrete compactness techniques, entropy-dissipation estimates and a new adaptation of the so-called duality estimates for parabolic equations in Laplacian form. I will also present numerical experiments illustrating the influence of the nonlocality in the system: on convergence properties, as an approximation of the local system and on the development of diffusive instabilities. This is a joint work with Antoine Zurek (UTC).

Single-cell RNA sequencing (scRNAseq) is a high-throughput technology quantifying gene expression at the single-cell level, for thousands of cells and tens of thousands of genes. A major statistical challenge in scRNAseq data analysis is to distinguish biological information from technical noise in order to compare conditions or tissues. Differential Expression Analysis (DEA) is usually performed with univariate two-sample tests and thus does not account for the multivariate aspect of scRNAseq data that carries information about gene dependencies and underlying regulatory networks and pathways. Applying multivariate two-sample tests would allow to perform Differential Transcriptome Analysis (DTA), to assess for the global similarity of the compared datasets.

We consider a one-dimensional simple random walk killed by quenched soft obstacles. The position of the obstacles is drawn according to a renewal process with a power-law increment distribution. In a previous work, we computed the large-time asymptotics of the quenched survival probability. In the present work we continue our study by describing the behaviour of the random walk conditioned to survive. We prove that with large probability, the walk quickly reaches a unique time-dependent optimal gap that is free from obstacle and gets localized there. We actually establish a dichotomy. If the renewal tail exponent is smaller than one then the walk hits the optimal gap and spends all of its remaining time inside, up to finitely many visits to the bottom of the gap.

L’optimisation stochastique englobe des méthodes permettant de minimiser une fonction de coût avec un caractère aléatoire, problème qui intervient souvent en machine learning et en particulier dans l’entraînement des réseaux de neurones. L'exemple le plus connu et le plus étudié d'une telle méthode est l'algorithme de la descente du gradient introduit par Robbins et Monro en 1951. Les algorithmes dits adaptatifs sont des extensions de cette descente de gradient stochastique classique qui visent à améliorer ses propriétés de convergence en déterminant automatiquement à chaque étape le taux d’apprentissage. Dans cette présentation on s’intéressera au comportement asymptotique des algorithmes de type RmsProp et Adagrad quand la fonction de coût est non-convexe.

TBA

Abstract

Les solutions des systèmes hyperboliques contiennent des discontinuités. Ces solutions faibles vérifient non seulement les EDP de départ, mais aussi une inégalité d'entropie qui agit comme un critère de sélection déterminant si une discontinuité est physique ou non. Il est très important d'obtenir une version discrète de ces inégalités d'entropie lorsqu'on approxime numériquement les solutions, sans quoi le schéma est susceptible de converger vers des solutions non physiques ou pire d'être instable. Obtenir une inégalité d'entropie discrète est en général un travail difficile, souvent inatteignable pour des schémas d'ordre élevé. Dans cet exposé, je présenterai une approche où ces inégalités sont obtenues a posteriori en minimisant une fonctionnelle bien choisie.