Estimation of partially known Gaussian graphical models with score-based structural priors

TL;DR

The paper tackles the problem of estimating partially known Gaussian graphical models by incorporating score-based priors over the adjacency graph. It develops a posterior-sampling framework based on annealed Langevin dynamics and augments it with a graph neural network that learns the score of the graph prior from data. The authors prove consistency of the constrained ML estimator and posterior concentration, and demonstrate substantial empirical gains over sparsity-focused baselines, especially when observational data are scarce or the underlying graphs are highly structured. The approach enables flexible integration of rich structural information into GGM structure learning and shows promise for applications where graphs are learned from related datasets. The combination of posterior sampling, annealing, and learned priors offers a principled and scalable path for partially known graph estimation in high dimensions.

Abstract

We propose a novel algorithm for the support estimation of partially known Gaussian graphical models that incorporates prior information about the underlying graph. In contrast to classical approaches that provide a point estimate based on a maximum likelihood or a maximum a posteriori criterion using (simple) priors on the precision matrix, we consider a prior on the graph and rely on annealed Langevin diffusion to generate samples from the posterior distribution. Since the Langevin sampler requires access to the score function of the underlying graph prior, we use graph neural networks to effectively estimate the score from a graph dataset (either available beforehand or generated from a known distribution). Numerical experiments demonstrate the benefits of our approach.

Figures (7)

• Figure 1: Illustration of our final algorithm [cf. Algorithm \ref{['alg:annealed_langevin']}]. The grey entries in ${\mathbf A}^{{\mathcal{O}}}_0$ are what we aim to estimate (i.e., ${\mathbf A}^{{\mathcal{U}}}$). If no entries of ${\mathbf A}_0$ are known, some can be estimated by bootstrapping ${\mathbf X}$, as shown in Section \ref{['sec:num_results']}. By combining the GGM observations ${\mathbf X}$ and the partially observed graph ${\mathbf A}^{{\mathcal{O}}}_0$, we compute the constrained ML estimator $\hat{\boldsymbol{\Theta}}$ by solving \ref{['eq:glasso']}. This encodes information about the likelihood of ${\mathbf A}$ given ${\mathbf X}$. To encode information about the prior $p({\mathbf A})$, we process the dataset ${\mathcal{A}}$ with a GNN (Section \ref{['subsec:gnn']}). Then, we can draw $M$ samples from the (approximate) posterior using a Langevin sampler and build any estimator with them, such as ${\hat{\mathbf A} }$ in \ref{['eq:a_est']} that approximates the posterior mean.
• Figure 2: F1 score of several methods using grid graphs with $40 \! \leq \! n \! \leq \! 50$ where (a) $10\%$ and (b) $20\%$ of the values in ${\mathbf a}$ are unknown.
• Figure 3: F1 score of several methods using (a) Barabási-Albert graphs with $|{\mathcal{U}}| = 0.1\dim({\mathbf a})$, and (b) ego-nets with $|{\mathcal{U}}| = 0.5\dim({\mathbf a})$.
• Figure 4: F1 score comparison when estimating ego-nets with no known values in ${\mathbf A}$. The values in parentheses correspond to the $p_m$ used to fix values from ${\hat{\mathbf A} }_\mathrm{boot}$ prior to using our method LPost.
• Figure 5: Fitting of $\lambda(k) = a\log(k)^2 + b\log(k) + c$ for the ego-nets dataset with $|{\mathcal{U}}|=0.5\dim({{\mathbf a}})$. The orange curve on the right is the one used at inference time.

Theorems & Definitions (3)

• proof
• proof
• proof