Residual connections are ubiquitous in deep learning, since besides residual networks and their variants, they are also present in Transformer architectures. The dynamic view of residual networks views them as analogous to a forward Euler scheme for an ordinary differential equation. We can then say that residual networks transport input points in ...
La méthode ICP est devenue une méthode couramment utilisée pour du recalage rigide de représentations surfaciques. Plusieurs améliorations furent publiées pour tenter de combler les lacunes de la méthode. Depuis 2013, le domaine du transport optimal a connu un regain de popularité grâce à la parallélisabilité du calcul de plans de transports. Nous ...
Supervised learning algorithms aims at identifying relationship between inputs and outputs thanks to training sets of couples (input, output). The most studied setting of supervised learning deals with high-dimensional inputs but low-dimensional outputs, as, for example, real numbers in the case of regression, and the values zero or one in the case...
Optimal transportation is a resource allocation problem present in fields such as economics, finance, physics or artificial intelligence. From a probabilistic point of view, the optimal transport cost endows the space of probability measures with a metric topology. In particular, this topology is equivalent to the weak topology of probability measu...
In this thesis we investigate some properties of solutions of L∞-variational and transport problems. This manuscript is divided into two parts. The first part, made up of Chapter 2 and Chapter 3, deals with a supremal variational problem. Supremal variational problems appeared for the first time in the late 60s in the pioneering works of Aronsson [...
A key challenge in Machine Learning (ML) is to design models able to learn efficiently from graphs, characterized by nodes with attributes and a prescribed structure encoding their relationships. Graph Representation Learning (GRL) aims to encode these two sources of heterogeneity into a vectorial graph embedding easing downstream tasks. In this fi...
Mixture models are relevant to represent several sub-populations inside a global population. In these models, the weights parameter accounts for the proportions of the different sub-populations that compose the global population. In this thesis, we develop new tools for the estimation of the weights parameter. Our developments are based on the noti...
In this thesis, we will propose mathematical models and methods based on partial differential equations to compress images and videos. In the first part, we will propose a mathematical criterion allowing us to locate the best pixels to keep within an image in the case where the inpainting method is the heat equation. To do so, we will write the com...
Optimal transport theory not only defines a distance between probability measures but also provides a geometric way to transport a set of points to another according to the principle of least effort. This dual aspect has left the door wide open for applications in domain adaptation, a subfield of statistical learning theory that takes into account ...
In various domains, certain phenomena that depend on time are represented by mathematical models. In particular the collective behavior of a group composed of a certain number of agents. In kinetic theory models, we study the position and velocity of each agent over time. The Boltzmann equation is one of the best-known equations in this domain. In ...
