Transductive and inductive adaptative inference for regression and density estimation.

Summary Adaptive, Inductive and Transductive Inference for Regression and Density Estimation (Pierre Alquier) The purpose of this thesis is to study the statistical properties of some learning algorithms in the case of regression and density estimation. It is divided into three parts. The first part consists in a generalization of Olivier Catoni's PAC-Bayesian theorems on classification to the case of regression with a general loss function. In the second part, we study more particularly the case of least squares regression and we propose a new variable selection algorithm. This method can be applied in particular to the case of a basis of orthonormal functions, and then leads to optimal convergence speeds, but also to the case of kernel type functions, it then leads to a variant of the so-called "support vector machines" (SVM) methods. The third part extends the results of the second part to the case of density estimation with quadratic loss.