[Paper Review] Learning Gaussian Graphical Models by symmetric parallel regression technique

Daniela De Canditiis, Armando Guardasole|arXiv (Cornell University)|Feb 8, 2019

Metabolomics and Mass Spectrometry StudiesBiochemistry, Genetics and Molecular Biology2 references3 citations

TL;DR

This paper proposes a symmetric parallel regression (SPR) technique for learning sparse Gaussian graphical models (GGMs) in high-dimensional settings, leveraging grouped-lasso penalized regression to estimate conditional independence structures. SPR achieves competitive performance in edge recovery and significantly reduces computational time compared to Graphical Lasso and standard parallel regression, especially under high-dimensional regimes (n << p).

ABSTRACT

In this contribution we deal with the problem of learning an undirected graph which encodes the conditional dependence relationship between variables of a complex system, given a set of observations of this system. This is a very central problem of modern data analysis and it comes out every time we want to investigate a deeper relationship between random variables, which is different from the classical dependence usually measured by the covariance. In particular, in this contribution we deal with the case of Gaussian Graphical Models (GGMs) for which the system of variables has a multivariate gaussian distribution. We study all the existing techniques for such a problem and propose a smart implementation of the symmetric parallel regression technique which turns out to be very competitive for learning sparse GGMs under high dimensional data regime.

Motivation & Objective

• Address the challenge of learning conditional independence structures in high-dimensional multivariate Gaussian data where the empirical covariance matrix is singular.
• Overcome limitations of traditional methods like Graphical Lasso and standard parallel regression in high-dimensional regimes (n ≪ p).
• Develop a computationally efficient and accurate algorithm for estimating the sparse precision matrix Θ = Σ⁻¹ in Gaussian graphical models.
• Demonstrate that symmetric parallel regression can achieve competitive performance in edge recovery while significantly reducing runtime compared to existing methods.

Proposed method

• Formulate the problem of learning a GGM as estimating the support of the inverse covariance matrix (precision matrix) using conditional independence relationships.
• Implement symmetric parallel regression (SPR) by solving p separate grouped-lasso penalized regression problems, one for each variable, using the remaining variables as predictors.
• Enforce symmetry in the estimated precision matrix by averaging the regression coefficients from each regression step, ensuring consistency with the true precision matrix structure.
• Use a fixed regularization parameter λ = log(p)/n, which is theoretically justified as optimal in high-dimensional settings.
• Apply the grouped-lasso penalty to promote sparsity in the estimated conditional dependence structure, favoring edge selection only when significant.
• Estimate the final precision matrix as the average of the coefficient matrices from all p regressions, ensuring symmetry and improved stability.

Experimental results

Research questions

• RQ1 How does symmetric parallel regression compare to Graphical Lasso and standard parallel regression in terms of edge recovery accuracy in high-dimensional GGMs?
• RQ2 Can symmetric parallel regression achieve competitive performance with significantly reduced computational cost in high-dimensional settings (n ≪ p)?
• RQ3 How robust is the SPR method across different graph topologies (e.g., chain, grid, star) under varying data regimes?
• RQ4 Does the symmetric averaging step in SPR improve estimation stability and accuracy compared to asymmetric regression approaches?
• RQ5 What is the impact of using λ = log(p)/n on the performance of SPR in high-dimensional sparse GGM learning?

Key findings

• SPR achieves competitive edge recovery accuracy compared to Graphical Lasso and standard parallel regression, with accuracy values close to the best-performing method across all tested graph types.
• SPR demonstrates significantly faster computational runtime than Graphical Lasso, which is known to be computationally intensive due to its iterative optimization procedure.
• In both not-severe (n ≈ p) and severe (n ≪ p) high-dimensional regimes, SPR maintains stable performance across different graph structures (chain, grid, star), indicating robustness.
• The symmetric averaging step in SPR improves estimation consistency and contributes to better stability in precision matrix estimation compared to non-symmetric alternatives.
• For the tested scenarios (p = 32, n = 32 and n = 16), SPR outperforms standard parallel regression in computational efficiency while maintaining comparable accuracy.
• The implementation of SPR is computationally efficient and scalable, making it a viable alternative to Graphical Lasso in high-dimensional, sparse GGM learning tasks.