Asymptotic Methods for High-Dimensional Estimation and Learning
I will present recent work on using asymptotic methods from probability theory and mean-field statistical physics to understand problems in high-dimensional estimation and learning. In particular, I will show (1) the exact characterization of a spectral method widely used in effective dimension reduction and exploratory data analysis; (2) the fundamental limits of solving the phase retrieval problem via linear programming; and (3) how to use scaling and mean-field limits to analyze iterative algorithms for nonconvex optimization. In all these problems, asymptotic methods clarify some of the fascinating phenomena, such as phase transitions, that emerge with high-dimensional data. They also lead to optimal designs that significantly outperform heuristic choices commonly used in practice.