High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

04/09/2022

∙

by   Tengyuan Liang, et al.

∙

0

∙

share

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. As a second contribution, we derive an explicit formula for the limiting overlap in terms of the signal-to-noise ratio and the injected noise in the diffusion. This uncovers a sharp phase transition – in one regime, the limiting overlap is strictly positive, while in the other, the injected noise overcomes the signal, and the limiting overlap is zero.