Sparse Inverse Covariance Selection via Alternating Linearization Methods

Gaussian graphical models are of great interest in statistical learning. Because the conditional independencies between different nodes correspond to zero entries in the inverse covariance matrix of the Gaussian distribution, one can learn the structure of the graph by estimating a sparse inverse covariance matrix from sample data, by solving a convex maximum likelihood problem with an $\ell_1$-regularization term. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem’s special structure; in particular, the subproblems solved in each iteration have closed-form solutions. Moreover, our algorithm obtains an $\epsilon$-optimal solution in $O(1/\epsilon)$ iterations. Numerical experiments on both synthetic and real data from gene association networks show that a practical version of this algorithm outperforms other competitive algorithms.

💡 Research Summary

The paper addresses the problem of estimating a sparse inverse covariance (precision) matrix, which is central to learning the structure of Gaussian graphical models. In such models, a zero entry in the precision matrix corresponds to conditional independence between two variables, so recovering the graph reduces to solving a convex maximum‑likelihood problem with an ℓ₁ penalty that promotes sparsity. Traditional solvers—such as interior‑point methods, QUIC, or ADMM‑based Graphical Lasso—must handle the non‑smooth ℓ₁ term together with the highly non‑linear log‑determinant term. Consequently, they either require expensive second‑order information or involve inner iterative loops, leading to high computational and memory costs, especially when the dimension p exceeds a few thousand.

The authors propose a first‑order algorithm based on Alternating Linearization (AL). The objective

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