ADMM Algorithm for Graphical Lasso with an $\ell_{\infty}$ Element-wise Norm Constraint

TL;DR

This paper introduces an ADMM-based algorithm with a continuation strategy to efficiently solve the Graphical Lasso problem under an $\,\ell_{\,\infty}$ element-wise norm constraint, relevant for high-dimensional covariance decomposition.

Contribution

It presents a novel ADMM algorithm with a continuation strategy specifically designed for Graphical Lasso with an $\,\ell_{\,\infty}$ constraint, enhancing computational efficiency.

Findings

Algorithm achieves faster convergence with the continuation strategy.

Effective in high-dimensional covariance decomposition tasks.

Provides a practical solution for constrained graphical model estimation.

Abstract

We consider the problem of Graphical lasso with an additional $\ell_{\infty}$ element-wise norm constraint on the precision matrix. This problem has applications in high-dimensional covariance decomposition such as in \citep{Janzamin-12}. We propose an ADMM algorithm to solve this problem. We also use a continuation strategy on the penalty parameter to have a fast implemenation of the algorithm.