Milica Tomacevic – Tosca
From Machine Learning To Finance : A view on Stochastic Gradient Algorithm
Victor Reutenauer – Tosca
Stochastic gradient algorithm are used for optimization problems or calibration of a model. We present shortly their theoretical and historical aspects, and then their application in two fields. First how it can be used to solve stochastic control problem like hedging of portfolio in incomplete market with application to indifference pricing. Then we present their application to machine learning and computer vision. We draw a parrallel between control stochastic problem and calibration of neural network for object and scene detection. We present also their usage for application to Kmean algorithm that is used in quantization methods as well as clustering of data for which we present indexation of media. Namely having a large database of photos to query into.
Description of the lack of compactness in some functional embeddings and application to PDEs
David Lafontaine, LJAD (Laboratoire Jean Alexandre Dieudonné) – Géometrie, Analyse et Dynamique
Solutions of a lot of partial differential equations can be obtained as minimizers of certain quantities. When dealing with a minimizing sequence in a compact set, we can easily extract a limit and prove the existence of a solution, but in infinite dimension, boundnessness is not sufficient to obtain compactness. Describing the lack of compactness we are dealing with, can permit to extract subsequences. More recently, this approach has been extended to dynamical problems, in particular by Kenig and Merle, with a remarkable success. As an example, we present a recent result we obtained concerning the large-time behavior of the non linear Schrödinger equation with a potential. More precisely, we show that the equation behaves in a linear way for large times.