High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models
This addresses signal recovery in high-dimensional statistics, providing theoretical insights into algorithmic behavior, but it appears incremental as it builds on existing spin glass methods.
The paper studies Langevin dynamics for signal recovery in spiked matrix models, deriving a path-wise characterization of overlap via CHSCK equations and uncovering a sharp phase transition where limiting overlap is positive or zero depending on signal-to-noise ratio.
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.