sparse graphical model estimation

This paper addresses statistical inference for multiple Gaussian graphical models sharing a common sparse precision matrix structure. We propose a debiased group graphical Lasso method that circumvents conventional incoherence-type conditions, yielding asymptotically unbiased estimators. Under moderate high-dimensionality ($p_n = o(n^{1/2})$), the method enables simultaneous hypothesis testing for nonzero entries across all group-specific precision matrices. Theoretically, the estimator achieves an $O_P(sqrt{log p / n})$ convergence rate in estimation error, enjoys model selection consistency, and exhibits asymptotic normality. Simulation studies demonstrate substantial improvements in statistical power and confidence interval coverage compared to existing approaches. Applied to real fMRI data, the method successfully identifies stable, cross-population brain functional connectivity patterns—thereby validating its statistical reliability and practical utility.

This paper addresses graph structure learning for time-varying mean Gaussian graphical models (GGMs), where conventional Graphical Lasso suffers from bias in precision matrix estimation due to its restrictive zero-mean assumption. We propose an iterative framework that jointly estimates the time-varying mean and sparse precision matrix. To our knowledge, this is the first approach integrating Bayesian inference with frequentist estimation for GGMs—introducing an Adaptive Targeted Adaptive Importance Sampling (ATAIS) mechanism that operates without prior assumptions on the mean, enabling robust co-estimation of mean dynamics and graph topology. The method synergistically combines importance sampling, regularized maximum likelihood estimation, and sparse optimization. Extensive experiments on multiple synthetic and real-world datasets demonstrate an average 23.6% improvement in F1-score over baselines, significantly mitigating spurious edge detection induced by mean shifts.

Regularization is often used in high-dimensional regression settings to generate a sparse model, which can save tremendous computing resources and identify predictors that are most strongly associated with the response. When the predictors can be represented by a Gaussian graphical model, the structure of the predictor graph can be exploited during regularization. Our proposed model exploits this underlying predictor graph structure by decomposing the estimated coefficient vector into a sum of latent variables that correspond to the sum of each node contribution to the coefficient vector. Regularization is then performed on the latent variables rather than on the coefficient vector directly. We use a penalty function that permits a clear user-defined trade-off between the L1 and L2 penalties and propose a novel proximal projection during optimization. Further, our implementation computes the projection operator for the intersection of selected groups, which conserves more computing resources compared to predictor duplication methods, especially for high-dimensional data. Through simulation, we evaluate the performance of our approach under different graph structures and node counts, and present results on real-world data. Results suggest that our method exhibits stable performance relative to other singly or doubly sparse graphical regression models.

This work addresses the tendency of Laplacian-constrained graphical models—such as the Laplacian-constrained Gaussian graphical model (LCGGM) and the Hüsler–Reiss model—to yield overly dense graphs during structure learning, which compromises interpretability and scalability. The paper introduces, for the first time, spectral graph sparsification as a post-processing step: without requiring additional hyperparameter tuning, it replaces the original Laplacian estimate with a spectrally approximated sparse Laplacian and refits the model. This approach effectively enhances both the sparsity and accuracy of the estimated graph structure. Integrating spectral graph theory, Laplacian-constrained Gaussian graphical models, extreme-value graphical models, and graph sparsification techniques, the method demonstrates superior performance on Erdős–Rényi and stochastic block model simulations and validates its practical utility on real-world data.

In high-dimensional Gaussian graphical models, a single global regularization parameter often fails to accommodate scale heterogeneity across nodes—as commonly observed in fMRI data—thereby limiting the accuracy of recovered graph structures. To address this, this work proposes LARGE, a novel method that, within the GLASSO framework, employs a block coordinate descent algorithm to jointly estimate node-specific regression coefficients and error variances, enabling adaptive learning of node-level regularization parameters. LARGE is the first approach to achieve node-wise adaptive regularization without requiring data standardization, significantly enhancing both the accuracy and stability of graph estimation. Extensive experiments demonstrate that LARGE consistently outperforms existing methods on both synthetic and real fMRI datasets, yielding more reliable brain functional connectivity networks.

This work addresses the excessive density of Gaussian graphical models learned under multivariate total positivity of order two (MTP2) constraints, which compromises interpretability and computational efficiency. To overcome this limitation, the authors propose Spectral-MTP2, a novel method that integrates spectral graph sparsification into the MTP2-constrained graph learning framework. Spectral-MTP2 automatically constructs a sparse subgraph that closely approximates the original dependency structure while strictly preserving the MTP2 property. The approach is tuning-free, scalable, and statistically consistent. Empirical evaluations on both synthetic and real-world datasets—including stock returns and gene expression profiles—demonstrate that Spectral-MTP2 substantially enhances graph sparsity and interpretability without sacrificing model fit, achieving performance comparable to that of dense MTP2 models.

This work addresses the joint estimation of multiple precision matrices in high-dimensional settings where they share a common sparsity pattern yet exhibit heterogeneous edge strengths. The authors propose Multiplicative Graphical Lasso (Mglasso), which decomposes each precision matrix as the Schur–Hadamard product of a shared structural matrix and a group-specific strength matrix. This decomposition uniquely disentangles graph topology from edge intensities, enabling simultaneous learning of structural consistency and strength heterogeneity within a unified framework that combines ℓ₁ and Frobenius norm penalties. The resulting penalized log-likelihood is efficiently optimized via an alternating direction method of multipliers (ADMM) coupled with gradient descent. Theoretical analysis establishes high-dimensional consistency and exact support recovery, while experiments demonstrate that Mglasso significantly outperforms Group Graphical Lasso in small-sample regimes, achieving superior model selection consistency and practical utility.

This work addresses the computational inefficiency in estimating precision matrices for high-dimensional fully positive Gaussian graphical models by proposing a generalized Bayesian framework based on the D-trace loss, which circumvents costly log-determinant computations. To induce sparsity, a spike-and-slab prior is incorporated. Three key techniques are introduced to enhance scalability and sampling efficiency in high dimensions: conditional independence–driven data augmentation, a fast matrix-normal sampler leveraging the Gram structure of the sample covariance, and an intertwining strategy that combines augmented and direct updates. Experimental results demonstrate that the proposed method achieves substantial computational acceleration on both synthetic and financial datasets while maintaining high estimation accuracy and superior graph structure recovery.

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.