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The recent development of compressed sensing has led to spectacular advancesin the understanding of sparse linear estimation problems as well as inalgorithms to solve them. It has also triggered a new wave of developments inthe related fields of generalized linear and bilinear inference problems, thathave very diverse applications in signal processing and are furthermore abuilding block of deep neural networks. These problems have in common that theycombine a linear mixing step and a nonlinear, probabilistic sensing step,producing indirect measurements of a signal of interest. Such a setting arisesin problems as different as medical or astronomical imaging, clustering, matrixcompletion or blind source separation. The aim of this thesis is to proposeefficient algorithms for this class of problems and to perform theirtheoretical analysis. To this end, it uses belief propagation, thanks to whichhigh-dimensional distributions can be sampled efficiently, thus making aBayesian approach to inference tractable. The resulting algorithms undergophase transitions just as physical systems do. These phase transitions can beanalyzed using the replica method, initially developed in statistical physicsof disordered systems. The analysis reveals phases in which inference is easy,hard or impossible. These phases correspond to different energy landscapes ofthe problem. The main contributions of this thesis can be divided into threecategories. 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 problemscalled blind sensor calibration. Third, a theoretical analysis of matrixcompressed sensing and of instabilities in Bayesian bilinear inferencealgorithms.

Statistical physics of linear and bilinear inference problems