Échantillonnage compressé et réduction de dimension pour l'apprentissage non supervisé

This thesis is motivated by the perspective of connecting compressed sensing and machine learning, and more particularly by the exploitation of compressed sensing techniques to reduce the cost of learning tasks. After a reminder of compressed sensing and a quick description of data analysis techniques in which similar ideas are exploited, we propose a framework for estimating probability density mixture parameters in which the training data is compressed into a fixed-size representation. We instantiate this framework on an isotropic Gaussian mixture model. This proof of concept suggests the existence of theoretical guarantees for reconstructing signals belonging to models beyond usual sparse models. We therefore study generalizations of stability results for linear inverse problems for very general models of signals. We propose conditions under which reconstruction guarantees can be given in a general framework. Finally, we consider an approximate nearest neighbor search problem exploiting signatures of the database vectors in order to save resources during the search step. In the case where the considered distance derives from a Mercer kernel, we propose to combine an explicit embedding of data followed by a signature computation step, which principally leads to a more accurate approximate search.