Statistical physics of linear and bilinear inference problems

The recent development of compressed sensing has led to spectacular advances in the understanding of sparse linear estimation problems as well as in algorithms to solve them. It has also triggered a new wave of developments in the related fields of generalized linear and bilinear inference problems, that have very diverse applications in signal processing and are furthermore a building block of deep neural networks. These problems have in common that they combine a linear mixing step and a nonlinear, probabilistic sensing step, producing indirect measurements of a signal of interest. Such a setting arises in problems as different as medical or astronomical imaging, clustering, matrix completion or blind source separation. The aim of this thesis is to propose efficient algorithms for this class of problems and to perform their theoretical analysis. To this end, it uses belief propagation, thanks to which high-dimensional distributions can be sampled efficiently, thus making a Bayesian approach to inference tractable. The resulting algorithms undergo phase transitions just as physical systems do. These phase transitions can be analyzed using the replica method, initially developed in statistical physics of disordered systems. The analysis reveals phases in which inference is easy, hard or impossible. These phases correspond to different energy landscapes of the problem. The main contributions of this thesis can be divided into three categories. First, the application of known algorithms to concrete problems: community detection, superposition codes and an innovative imaging system. Second, a new, efficient message-passing algorithm for a class of problems called blind sensor calibration. Third, a theoretical analysis of matrix compressed sensing and of instabilities in Bayesian bilinear inference algorithms.Comment: Phd thesi

Mean-field methods and algorithmic perspectives for high-dimensional machine learning

The main difficulty that arises in the analysis of most machine learning algorithms is to handle, analytically and numerically, a large number of interacting random variables. In this Ph.D manuscript, we revisit an approach based on the tools of statistical physics of disordered systems. Developed through a rich literature, they have been precisely designed to infer the macroscopic behavior of a large number of particles from their microscopic interactions. At the heart of this work, we strongly capitalize on the deep connection between the replica method and message passing algorithms in order to shed light on the phase diagrams of various theoretical models, with an emphasis on the potential differences between statistical and algorithmic thresholds. We essentially focus on synthetic tasks and data generated in the teacher-student paradigm. In particular, we apply these mean-field methods to the Bayes-optimal analysis of committee machines, to the worst-case analysis of Rademacher generalization bounds for perceptrons, and to empirical risk minimization in the context of generalized linear models. Finally, we develop a framework to analyze estimation models with structured prior informations, produced for instance by deep neural networks based generative models with random weights.Comment: Ph.D manuscrip