we discuss classification problems in high dimension. We study classification problems using three classical notions: complexity of decision boundary, noise, and margin. We demonstrate that under suitable conditions on the decision boundary, classification problems can be very efficiently approximated, even in high dimensions. If a margin condition is assumed, then arbitrary fast approximation rates can be achieved, despite the problem being high-dimensional and discontinuous. We extend the approximation results ta learning results and show close ta optimal learning rates for empirical risk minimization in high dimensional classification. [-]

The aim of this talk is to present a new variation formulation of the time-dependent many-body electronic Schrödinger equation with Coulombic singularities. More precisely, its solution can actually be expressed as the solution of a global space-time quadratic minimization problem that proves to be useful for several tasks:
1) first, it is amenable to Galerkin time-space discretization schemes, using an appropriate least-square formulation
2) it enables to yield a new variational principle for the construction dynamical low-rank approximations, that is different from the classical Dirac-Frenkel variational principle
3) it enables to obtain fully certified a posteriori error estimators between the exact solution and approximate solutions.
The present analysis can be applied to the electronic many-body time-dependent Schrödinger equation with an arbitrary number of electrons and interaction potentials with Coulomb singularities.[-]

Modern machine learning architectures often embed their inputs into a lower-dimensional latent space before generating a final output. A vast set of empirical results---and some emerging theory---predicts that these lower-dimensional codes often are highly structured, capturing lower-dimensional variation in the data. Based on this observation, in this talk I will describe efforts in my group to develop lightweight algorithms that navigate, restructure, and reshape learned latent spaces. Along the way, I will consider a variety of practical problems in machine learning, including low-rank adaptation of large models, regularization to promote local latent structure, and efficient training/evaluation of generative models.[-]

The development of a suitable, efficient and accurate numerical method to solve wave problems is encountered in many academic and industrial applications. The Boundary Integral Equation (BIE) technique, whose discretization is known as the Boundary Element Method (BEM), is an appealing alternative to classical domain method because it allows to handle problems defined on the exterior of bounded domains as easily as those defined in the interior, without the introduction of an artificial boundary to truncate the computational domain. Very recently, an Isogeometric Analysis based Boundary Element Method (IgA-BEM) has been proposed in literature for the numerical solution of frequency-domain (Helmholtz) wave problems on 3D domains admitting a multi-patch representation of the boundary surface. While being powerful and applicable to many situations, this approach shares with standard BEMs a disadvantage which can easily become significant in the 3D setting. Indeed, when the required accuracy is increased, it can soon lead to large dense linear systems, whose numerical solution requires huge memory, resulting also in important computational cost. Recently the development of fast H-matrix based direct and iterative solvers for oscillatory kernels, as the Helmholtz one, has been studied. Here, we investigate the effectiveness of the H-matrix technique, along with a suitable GMRES iterative solver, when used in the context of multi-patch IgA-BEM.[-]

Tensor methods have emerged as an indispensable tool for the numerical solution of high-dimensional problems in computational science, and in particular problems arising in stochastic and parametric analyses. In many practical situations, the approximation of functions of multiple parameters (or random variables) is made computationally tractable by using low-rank tensor formats. Here, we present some results on rank-structured approximations and we discuss the connection between best approximation problems in tree-based low-rank formats and the problem of finding optimal low-dimensional subspaces for the projection of a tensor. Then, we present constructive algorithms that adopt a subspace point of view for the computation of sub-optimal low-rank approximations with respect to a given norm. These algorithms are based on the construction of sequences of suboptimal but nested subspaces.

Keywords: high dimensional problems - tensor numerical methods - projection-based model order reduction - low-rank tensor formats - greedy algorithms - proper generalized decomposition - uncertainty quantification - parametric equations[-]

Many problems in computational and data science require the approximation of high-dimensional functions. Examples of such problems can be found in physics, stochastic analysis, statistics, machine learning or uncertainty quantification. The approximation of high-dimensional functions requires the introduction of approximation tools that capture specific features of these functions.
In this lecture, we will give an introduction to tree tensor networks (TNs), or tree-based tensor formats. In part I, we will present some general notions about tensors, tensor ranks, tensor formats and tensorization of vectors and functions. Then in part II, we will introduce approximation tools based on TNs, present results on the approximation power (or expressivity) of TNs and discuss the role of tensorization and architecture of TNs. Finally in part III, we will present algorithms for computing with TNs. This includes algorithms for tensor truncation, for the solution of optimization problems, for learning functions from samples...[-]

Lorsque l'on évoque Darwin et la théorie de l'évolution, on ne pense pas aux mathématiques. Pourtant dès que l'on s'intéresse aux mécanismes de la sélection naturelle, au hasard de la reproduction et au rôle des mutations, il est indispensable de les utiliser.
Après une introduction historique aux idées de Darwin sur l'évolution des espèces, nous expliquons l'impact de sa théorie et de ses réflexions sur la communauté scientifique et l'influence qu'il a eue sur la modélisation mathématique des dynamiques de population ou de la génétique des populations. Nous développons quelques exemples d'objets mathématiques, tels les processus de branchement, qui permettent de prédire le futur d'une population (son extinction, sa diversité…) ou au contraire d'en connaître le passé biologique (l'ancêtre commun d'un groupe d'individus par exemple). L'introduction du hasard dans la modélisation des questions liées à la biodiversité et à l'évolution est fondamentale. Elle permet de prendre en compte les variabilités individuelles et de mieux comprendre l'impact des facteurs écologiques et génétiques sur l'évolution des espèces.
Ces idées seront illustrées par des exemples issus de travaux récents développés entre mathématiciens et biologistes.[-]