High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

TL;DR

This paper analyzes the behavior of Langevin dynamics in high-dimensional spiked matrix models, providing a detailed characterization of the overlap with the planted signal and revealing a phase transition based on noise levels.

Contribution

It introduces a path-wise analysis of Langevin dynamics in spiked matrix models and derives explicit formulas for the limiting overlap, highlighting a sharp phase transition.

Findings

Explicit formula for the limiting overlap in terms of SNR and noise

Identification of a sharp phase transition in the recovery performance

Characterization of the overlap via integro-differential equations

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 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.