Phase transitions in high-dimensional generalized linear models

I will present a framework for rigorously establishing the information-theoretic limits in high-dimensional generalized linear models (GLMs). The GLM includes as special cases plethora of important models in signal processing (compressed-sensing, phase retrieval etc), communications, but also in learning such as the famous perceptron neural network. Many instances of GLMs have been analyzed in the statistical physics literature, in particular thanks to the heuristic replica method developed in the context of spin glasses. I will discuss a recent technique called « adaptive interpolation method » that allows to vindicate the statistical approach in a unified manner, as well as recent findings about the rich algorithmic behaviors encountered in such models.