Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing

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.

• The paper introduces a polynomial-time algorithm to recover sparse GGMs from a non-mixing Glauber trajectory using minimal assumptions.
• It employs innovative local window patterns like 'iiji' for robust diagonal estimation and unbiased edge testing despite temporal dependencies.
• The work establishes theoretical sample complexity bounds and compares favorably to prior mixing-based methods in high-dimensional settings.

Efficient Structure Learning of Sparse Gaussian Graphical Models from Glauber Dynamics without Mixing

Problem Formulation

This work addresses the structure learning problem for high-dimensional Gaussian Graphical Models (GGMs), particularly under realistic temporal data regimes. Unlike the classical i.i.d. setting, many scientific applications generate temporally correlated data, frequently via local update dynamics such as Glauber (single-site Gibbs) sampling. The goal is to recover the conditional independence graph—the support of the precision matrix $\Theta$—from a single trajectory of continuous-time Glauber dynamics, without relying on chain mixing, or additional regularity beyond sparsity and minimum edge strength.

Summary of Contributions

The authors introduce a polynomial-time algorithm to recover the structure (and parameters) of $(\alpha,d)$-sparse GGMs from a single non-mixing Glauber trajectory. The trajectory length required for recovery is:

$$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$

where $n$ is the number of variables, $d$ the maximal degree, $\alpha$ the minimum normalized edge strength, and $\delta$ the failure probability. Crucially, the guarantee holds irrespective of the mixing time, which can be arbitrarily large, as the algorithm does not require the dynamical Markov chain to reach stationarity.

Key technical innovations include:

• Diagonal Estimation and Normalization: Conditional variances (diagonal entries of $\Theta$) are accurately estimated (via robust quantile methods) using local patterns in the trajectory, enabling coordinate-wise normalization to handle arbitrarily ill-conditioned matrices.
• Local Edge Testing: For each candidate edge $(i,j)$, the algorithm identifies short trajectory windows with prescribed update patterns ("iiji") that isolate pairwise influences and break problematic dependencies inherent in simpler ("iji") patterns.
• Robust Aggregation: Median-based estimators are applied to the ratio statistics extracted from local windows, mitigating contamination from unobserved neighbor updates and temporal dependencies.

Theoretical results also include:

• Parameter Learning: Recovery of all entries of $\Theta$ up to multiplicative $(\alpha,d)$0 error, with a trajectory length scaling as $(\alpha,d)$1.
• Mixing-Aware Algorithm: When rapid mixing can be assumed, a more sample-efficient algorithm using shorter "iji" patterns is derived, yielding a trajectory length of $(\alpha,d)$2.

An information-theoretic lower bound (via Fano's inequality and KL-divergence analysis) establishes that $(\alpha,d)$3 Glauber updates are necessary for structure recovery, matching the logarithmic dependence but leaving a polynomial gap in $(\alpha,d)$4, $(\alpha,d)$5.

Algorithmic Details

Diagonal Estimation

Diagonal entries $(\alpha,d)$6 are estimated by identifying "ii" update windows in the trajectory, such that no hidden neighbor of $(\alpha,d)$7 updates within the window. On these windows, differences in $(\alpha,d)$8 over successive updates yield Gaussian samples whose variance is tightly linked to $(\alpha,d)$9. Robust variance estimators, tolerant to contamination from hidden updates, deliver accurate diagonal estimates.

Local Edge Testing

For off-diagonal entries, naive "iji" update windows introduce statistical dependencies that invalidate mean-zero assumptions and contaminate ratio statistics. The algorithm circumvents this by requiring two successive $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$0 updates before the $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$1 update ("iiji" pattern), ensuring that noise terms in the numerator and denominator are independent. The derived ratio statistic is unbiased for $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$2, conditioned on the window being clean (free from neighbor updates).

Robust Median-Based Estimation

Temporal dependence and hidden neighbor contamination are addressed by accepting all windows that exhibit the visible update pattern and passing ratio statistics through a median-based aggregation procedure. Martingale concentration inequalities are leveraged to guarantee robust estimation under sequential contamination, yielding tight control on estimation error.

Improved Mixing-Based Regime

When mixing time is small, accepted windows are spaced by at least $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$3, mimicking near-stationarity; here, traditional covariance-based statistics suffice, and fewer trajectory samples are needed. This algorithm achieves the information-theoretic optimum for i.i.d. data under rapid mixing.

Numerical and Practical Remarks

The algorithm achieves strong theoretical guarantees but with large polylogarithmic constants in the trajectory length required. The authors explicitly note that these constants render practical deployment infeasible for real-world datasets, suggesting that tradeoffs in window length and contamination rate are fundamental to the mixing-free setting. Practical algorithms (with manageable sample sizes) for this regime remain an open avenue for further research.

Comparison to Prior Work

Prior work such as "Structure learning in Gaussian graphical models from Glauber dynamics" [tirukkonda2025structure] also addresses similar problems but relies on additional spectral and marginal regularity assumptions (such as explicit bounds on $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$4, and $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$5). Those settings typically guarantee rapid mixing, reducing the problem to approximate i.i.d. sampling. The present algorithm removes these requirements, relying solely on sparsity and minimum edge strength, thus applying to arbitrary ill-conditioned precision matrices. Moreover, the sample complexity for mixing-based algorithms here improves dependence on $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$6 and $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$7 compared to earlier results.

Implications and Future Directions

Practical Implications

This result extends the frontier of structure learning for GGMs to regimes where i.i.d. sampling is unavailable or mixing times are prohibitively large. Applications in neuroscience, genomics, finance, and climate science—where data arises from persistent Markov processes—may benefit from principled structure estimation even in the presence of severe temporal dependence.

Theoretical Implications

The mixing-free estimation algorithm demonstrates that polynomial-time structure recovery is achievable with only sparsity and minimum edge strength constraints, circumventing mixing-based limitations. The persistent gap between the upper and lower bounds in the polynomial parameters, especially in the dependence on $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$8 and $$T = \Omega\left( \frac{d^3 \log(n/\delta)}{\alpha^5} \right)$$9, invites further research into optimality and algorithmic improvements.

Speculative Directions

• Reducing Constants: Developing algorithms that achieve practically feasible trajectory lengths, possibly by refining the tradeoff in contamination rates, improving pattern detection, or incorporating adaptive window selection.
• General Dynamics: Extending to more complex Markov dynamics beyond Glauber, including reversible chains or Metropolis updates, especially in non-Gaussian settings.
• Robustness to Model Misspecification: Addressing scenarios where the trajectory is observed with noise, non-Gaussian perturbations, or latent structure.

Conclusion

This work achieves efficient structure and parameter learning for sparse GGMs from a single trajectory of non-mixing Glauber dynamics, under minimal assumptions. It sets a new benchmark for temporal structure learning, validating polynomial-time recovery in settings far outside rapid mixing regimes. The results broaden the applicability of graphical structure learning methods and motivate further study into practically efficient algorithms and optimal sample complexities in temporally correlated and high-dimensional statistical models.

Reference: "Learning Gaussian Graphical Models from a Glauber Trajectory Without Mixing" (2606.31230)