High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

We study Langevin dynamics for recovering the planted signal in the spiked matrix model. We provide a path-wise characterization of the overlap between the output of the Langevin algorithm and the planted signal. This overlap is characterized in terms of a self-consistent system of integro-differential equations, usually referred to as the Crisanti-Horner-Sommers-Cugliandolo-Kurchan (CHSCK) equations in the spin glass literature.

April 2022 · Tengyuan Liang, Subhabrata Sen, Pragya Sur