# High-Dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

Tengyuan Liang, Subhabrata Sen, Pragya Sur
### Abstract

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 integrodifferential 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. As an upshot, 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.

Publication

Information and Inference: A Journal of the IMA