On the Equivalence of Gaussian Graphical Models Defined on Complete Bipartite Graphs

2025/05/05 by Mehdi Molkaraie, Molkaraie, Mehdi
Computer Science · #Computation (stat.CO) #Computational Complexity (cs.CC) #Data Management and Algorithms #FOS: Computer and information sciences #Graph Theory and Algorithms #Information Theory (cs.IT)

paper · pdf · doi:10.48550/arxiv.2505.02384

openalex publication_date 2025/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces two Gaussian graphical models defined on complete bipartite graphs. We show that the determinants of the precision matrices associated with the models are equal up to scale, where the scale factor only depends on model parameters. In this context, we will introduce a notion of ``equivalence" between the two Gaussian graphical models. This equivalence has two key applications: first, it can significantly reduce the complexity of computing the exact value of the determinant, and second, it enables the derivation of closed-form expressions for the determinants in certain special cases.

Citations

• The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices
• Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions
• A Randomized Algorithm for Approximating the Log Determinant of a\n Symmetric Positive Definite Matrix
• Adaptive Thresholding for Sparse Covariance Matrix Estimation
• Codes on graphs: Duality and MacWilliams identities
• Approximation of the determinant of large sparse symmetric positive definite matrices

Related

• The Exact Determinant of a Specific Class of Sparse Positive Definite Matrices
• On perfectness in Gaussian graphical models
• Duality for Continuous Graphical Models
• Positivity for Gaussian graphical models
• Nested Covariance Determinants and Restricted Trek Separation in\n Gaussian Graphical Models