Topological Exploration of High-Dimensional Empirical Risk Landscapes: General Approach, and Applications to Phase Retrieval

Abstract

How is the landscape of a high-dimensional learning problem organized, and how many minima, saddles and maxima does it contain? Building on tools from the statistical physics of random landscapes (the Kac-Rice formula), we develop a general and tractable framework to count and characterize the critical points of empirical risk landscapes, and use it to draw complete topological phase diagrams for the phase retrieval problem.

Tony Bonnaire

CNRS AI Research Scientist