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Leçons J.-L. Lions 2021 - 16 11 2021 12h - Mini-cours 1 : D. Slepčev
Mardi 16 novembre 2021 — 12h00-13h15
Leçons Jacques-Louis Lions 2021 - Mini-cours 1
Dejan Slepčev (Université Carnegie Mellon, Pittsburgh)
Variational problems and PDE on random structures : analysis and applications to data science
Le mini-cours sera donné dans la salle du séminaire du Laboratoire Jacques-Louis Lions :
Campus Jussieu, Sorbonne Université, 4 place Jussieu, Paris 5ème
barre 15-16, 3ème étage, salle 09 (15-16-3-09) ;
il sera retransmis en temps réel par Zoom.
Résumé du mini-cours
Many machine learning tasks, such as clustering, regression, classification, and dimensionality reduction are commonly described as optimization problems. Namely these tasks are modeled by introducing functionals (defined using the available random sample) which specify the desired properties of the object sought. While the data are often high dimensional, they usually have an intrinsic low-dimensional structure that makes the learning tasks feasible. The intrinsic geometric structures is often encoded by a graph created by connecting the nearby data points. We will introduce mathematical tools used to study variational problems and PDE-based models posed on random data samples. In particular we will discuss the passage from discrete optimization problems on random samples to their continuum limits. This will be used to establish asymptotic consistency of several important machine learning algorithms.
We will cover the basic elements of the background material on calculus of variations and optimal transportation. Furthermore we will develop connections to nonlocal functionals which serve as intermediate objects between the discrete functionals and their continuum limits. We will also consider approaches based on dynamics on graphs and connect these with the evolution equations describing the continuum limits.
Les Leçons Jacques-Louis Lions 2021 consistent en :
— un mini-cours
Variational problems and PDE on random structures : analysis and applications to data science
3 séances, mardi 16, mercredi 17 et jeudi 18 novembre 2021 de 12h à 13h15,
— et un colloquium
Machine learning meets calculus of variations
vendredi 19 novembre 2021 de 14h à 15h.
Plus d’information sur la page
https://www.ljll.math.upmc.fr/lecons-jacques-louis-lions-2021-dejan-slepcev