High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

Tengyuan Liang; Subhabrata Sen; Pragya Sur

doi:10.1093/imaiai/iaad042

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arXiv · 2204.04476

High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models

Tengyuan Liang·Subhabrata Sen·Pragya Sur

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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.

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BibTeX RIS

Tengyuan Liang, Subhabrata Sen, Pragya Sur. 2022-04-09. High-dimensional Asymptotics of Langevin Dynamics in Spiked Matrix Models. https://doi.org/10.1093/imaiai%2Fiaad042

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