Theoretical study of some statistical procedures applied to complex data.

Summary The main part of this thesis focuses on developing the theoretical and algorithmic aspects of three distinct statistical procedures. The first problem is the completion of binary matrices. We propose an estimator based on a pseudo-Bayesian variational approximation using a loss function different from those used previously. We can compute non-asymptotic bounds on the integrated risk. The proposed estimator is much faster to compute than an MCMC type estimate and we show on examples that it is efficient in practice. The second problem is the study of the theoretical properties of the empirical risk minimizer for lipschitzian loss functions. We can then apply the main results on logistic regression with SLOPE penalization and on matrix completion. The third chapter develops an Expectation-Propagation approximation when the likelihood is not explicit. We then use the ABC approximation in a second step. This procedure can be applied to many models and is much more accurate and fast. It is applied as an example on a model of spatial extremes.