Estimation of multiple precision matrices under shared support with heterogeneous edge strengths

2026-07-26

AI Summary

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.

Abstract

Estimating multiple precision matrices in high-dimension presents significant challenges, particularly when distinct datasets share a common conditional dependency structure but exhibit population-specific interaction strengths. We address this problem by introducing the Multiplicative Graphical Lasso (Mglasso), a method for jointly estimating precision matrices across multiple Gaussian graphical models under a shared sparsity constraint. Each precision matrix is decomposed as a Schur-Hadamard product of a shared structural matrix $\boldsymbolΘ$, which encodes the common conditional independence graph, and a population-specific matrix $\boldsymbolΓ_{l}$, which captures variation in edge strengths across populations. We optimize a penalized log-likelihood that utilizes an $\ell_1$-penalty to enforce common sparsity and a Frobenius norm penalty to regulate population-specific variations. The optimization is efficiently performed using the Alternating Direction Method of Multipliers (ADMM) algorithm integrated with gradient descent. Theoretically, we establish the local strict convexity of the objective function and provide rigorous high-dimensional consistency guarantees, including supremum norm error bounds and exact support recovery under sub-Gaussian tail conditions. Extensive simulations show superior model selection consistency at smaller sample sizes compared to the benchmark Group Graphical Lasso (GGL). Finally, the method's practical utility is further validated through real-world applications.

Problem

• precision matrix
• shared sparsity
• heterogeneous edge strengths
• Gaussian graphical model
• high-dimensional estimation

Innovation

• Multiplicative Graphical Lasso
• shared sparsity
• precision matrix estimation
• Schur-Hadamard product
• high-dimensional consistency