Mixing-Free and Signal-Optimal Learning of Gaussian Graphical Models from Glauber Dynamics

2026-07-20

AI Summary

This work addresses the problem of accurately recovering the structure of Gaussian graphical models from a single, non-stationary, dependent trajectory generated by Glauber dynamics, without relying on conventional assumptions such as Markov chain mixing time, spectral gap, or stationarity. To this end, the authors propose two update-sequence-based algorithms. The first recovers Gaussian innovations via least-squares regression combined with update pattern counting, requiring only Õ(pd²/κ²) updates and exhibiting merely logarithmic dependence on the local condition number. The second algorithm eliminates dependence on the condition number entirely at the cost of Õ(pd⁴/κ²) updates. Both methods operate from arbitrary initial states and, for the first time, achieve information-theoretically optimal dependence on the accuracy parameter κ⁻².

Abstract

Gaussian graphical model selection is usually studied under independent sampling, but in many applications the data arise as a single trajectory of a dependent stochastic process. We study exact recovery of the graph from one trajectory of random-scan Gaussian Glauber dynamics. Existing techniques for this problem either inherit the mixing time of the chain, which can be super-polynomial in the dimension $p$ without strong assumptions, or are suboptimal in the minimum normalized edge strength $κ$. We propose two algorithms that are mixing-free and attain the $κ^{-2}$ dependence of the information-theoretic lower bounds. Both instantiate a shared dueling-neighborhood search meta-algorithm with a local statistic built directly from the update sequence. The first fits a least-squares regression at the updates of each node and recovers the graph from $\widetilde O(pd^{2}/κ^{2})$ updates, where $d$ is the maximum degree. This algorithm's data requirement depends on a local conditioning quantity, but only logarithmically and is provably optimal even when the underlying chain mixes slowly. The second algorithm is based on counting occurences of a specific update pattern and requires $\widetilde O(pd^{4}/κ^{2})$ updates, with no dependence on any condition number. The central technical challenge is that both statistics are built from dependent, non-stationary observations. Our analysis tackles this by demonstrating how to extract fresh Gaussian innovations from the update sequence, which yields mixing-free control of appropriate quantities. Neither the algorithms nor their analyses invoke stationarity, a spectral gap, or mixing conditions, and all guarantees hold from an arbitrary initialization.

Problem

• Gaussian graphical models
• Glauber dynamics
• graph recovery
• dependent data
• mixing time

Innovation

• mixing-free
• Gaussian graphical models
• Glauber dynamics
• exact recovery
• non-stationary observations