Mixing times of Langevin dynamics for spiked matrix models

TL;DR

This paper studies the mixing times of Langevin dynamics in spiked matrix models, revealing a phase transition at a critical inverse temperature and showing that certain initializations enable rapid mixing even in low-temperature regimes.

Contribution

It demonstrates that for large signal-to-noise ratios, the mixing time sharply transitions at a critical point and that specific initializations can bypass exponential slowdowns, providing a detailed metastability analysis.

Findings

Mixing time is logarithmic in N for high signal-to-noise ratios.

Initialization from the uniform spherical prior leads to fast mixing in low-temperature regimes.

The exponential rate of mixing time is characterized by the difference in free energies of spiked and null models.

Abstract

We investigate the Langevin dynamics for Wigner matrices with a spherical spike, in the regime where the signal-to-noise ratio $\theta$ is large, but order one. For large, order-$1$, signal-to-noise, the (worst-case) mixing time undergoes a sharp transition around the critical inverse temperature $\beta_c(\theta) = \frac{1}{\theta}$. Namely, if $\beta = \alpha/\theta$, and $\alpha<1$ then at large $\theta$ the mixing time is $O(\log N)$, and if $\alpha>1$ it is exponential in $N$. We show that initialized from the uniform-at-random spherical prior, however, the mixing time in the low-temperature $\alpha>1$ regime circumvents the exponential bottleneck and the mixing time is $O(\log N)$. In fact, this fast mixing holds for any initialization that is symmetric with respect to the top eigenvector of the spiked matrix. Using this, we are able to show a low-temperature metastability picture, pinning down the exact exponential rate of the (worst-case initialization) mixing time for low temperatures, showing it is given by the difference of the free energies of the spiked and null models.