Accelerated Random-Sweep Gibbs Sampling for Gaussian Graphical Models via Dual Normal Factor Graphs

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. For every fixed precision matrix and deterministic initialization, the first algorithm fits a least-squares regression at the updates of each node and has pointwise recovery horizon $\widetilde O(pd^{2}/κ^{2})$, where $d$ is the maximum degree. Its horizon depends logarithmically on a local conditioning quantity and on the initialization potential. 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.

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.

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.