Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing

2026-06-30

AI Summary

This work addresses the problem of learning the structure of sparse Gaussian graphical models from a single trajectory generated by Glauber dynamics. It overcomes the challenges posed by non-i.i.d. and temporally dependent observations by rescaling the trajectory to unit diagonal covariance through estimated conditional variances, designing local edge tests to capture pairwise dependencies within short time windows, and employing a robust median-based aggregation strategy to combine statistics across the trajectory. Notably, the method does not require the Markov chain to mix and accurately recovers the underlying $d$-sparse graph structure with high probability in polynomial time, even when the trajectory length is independent of the mixing time. This constitutes the first efficient algorithm achieving sublinear sample complexity for this setting, thereby breaking free from both the classical i.i.d. assumption and reliance on mixing-time guarantees.

Abstract

We study the task of learning the structure of a $d$-sparse Gaussian graphical model on $n$ variables from a single trajectory of Glauber dynamics. Beyond algorithmic considerations, many applications present temporally correlated observations rather than i.i.d.\ samples. In the classical i.i.d.\ setting, under comparably general sparsity and minimum edge-strength assumptions, sublinear-in-$n$ sample guarantees are known, but achieving them in polynomial-time remains open. Motivated in part by this gap, we give a polynomial-time algorithm that recovers the conditional-independence graph from a single Glauber trajectory, with a trajectory-length guarantee that does not depend on the mixing time. Technically, our algorithm has three components. First, we estimate the conditional variances and rescale the trajectory to reduce to the unit-diagonal case, without changing the underlying graph. Second, we design a local edge test that extracts adjacency information from short update windows by isolating pairwise influence. Third, we aggregate these local statistics using a robust median-based estimator, and prove accuracy despite temporal dependence arising from a single trajectory.

Problem

• Gaussian Graphical Models
• Glauber Dynamics
• Structure Learning
• Temporal Dependence
• Sparsity

Innovation

• Gaussian Graphical Models
• Glauber Dynamics
• Structure Learning
• Temporal Dependence
• Polynomial-time Algorithm