Cette thèse propose des avancées théoriques et algorithmiques pour les relaxations semidéfinies positives et leurs applications en science des données. Ces relaxations, dites de Lasserre, fondées sur la substitution aux mesures boréliennes de leurs moments trigonométriques, permettent de résoudre des problèmes de super-résolution sans discrétisatio...
This thesis proposes theoretical and algorithmic advances for positive semi-definite relaxations and their applications in data science. These so-called Lasserre’s hierarchies allow one to solve super-resolution problems without resorting to spatial discretization, by replacing measures with their trigonometric moments. However, they require the re...
Topological data analysis (TDA) allows one to extract rich information from structured data (such as graphs or time series) that occurs in modern machine learning problems. This information will be represented as descriptors such as persistence diagrams, which can be described as point measures supported on a half-plane. While persistence diagrams ...
Analyse de données géométriques, au delà des convolutionsPour modéliser des interactions entre points, une méthode simple est de se reposer sur des sommes pondérées communément appelées "convolutions". Au cours de la dernière décennie, cette opération est devenue la brique de construction essentielle à la révolution du "deep learning". Le produit d...
The goal of this thesis is to develop new numerical methods to address inverse problems using optimal transport. Inverse problems appear in many disciplines such as astronomy, geophysics or medical imaging, but also in fields closer to the focus of this thesis, namely computer vision, computer graphics, and machine learning. They are difficult prob...
This thesis focuses on the approximation for the 2-Wasserstein metric of probability measures by structured measures. The set of structured measures under consideration is made of consistent discretizations of measures carried by a smooth curve with a bounded speed and acceleration. We compare two different types of approximations of the curve: pie...
L'étude des surfaces continentales constitue un enjeu majeur à l'échelle mondiale pour le suivi et la gestion des territoires, notamment en matière de répartition entre l'expansion urbaine, terres agricoles et espaces naturels. Dans ce contexte, les cartes d'OCcupation des Sols (OCS) caractérisant la couverture biophysique des terres émergées sont ...
The world we live in is constantly under observation. Many areas such as offshore zones, deserts, agricultural land and cities are monitored. This monitoring is done throughout remote sensing satellites or cameras mounted on aircrafts. However, because of many technological and financial constraints, the development of imaging sensors with high acc...
This thesis focuses on Incompressible Optimal Transport, a minimization problem introduced by Brenier in the late 80's, aiming at describing the evolution of an incompressible and inviscid fluid in a Lagrangian way , i.e. by prescribing the state of the fluid at the initial and final times and by minimizing some functional among the set of admissib...
L'espace de Wasserstein est l'ensemble des mesures de probabilité définies sur un domaine fixé et muni de la distance de Wasserstein quadratique. Dans ce travail, nous étudions des problèmes variationnels dans lesquels les inconnues sont des applications à valeurs dans l'espace de Wasserstein.Quand l'espace de départ est un segment, c'est-à-dire qu...
