# High-dimensional asymptotics of Langevin dynamics in spiked matrix models

arxiv（2023）

摘要

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 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.

更多查看译文

关键词

langevin dynamics,high-dimensional

AI 理解论文

溯源树

样例

生成溯源树，研究论文发展脉络

Chat Paper

正在生成论文摘要