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 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.