An optimal randomized parallel algorithm for finding connected components in a graph
SIAM Journal on Computing
Convex Optimization
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
The Journal of Machine Learning Research
Smooth Optimization Approach for Sparse Covariance Selection
SIAM Journal on Optimization
Adaptive First-Order Methods for General Sparse Inverse Covariance Selection
SIAM Journal on Matrix Analysis and Applications
High-dimensional Covariance Estimation Based On Gaussian Graphical Models
The Journal of Machine Learning Research
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We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter λ. Suppose the sample covariance graph formed by thresholding the entries of the sample covariance matrix at λ is decomposed into connected components. We show that the vertex-partition induced by the connected components of the thresholded sample covariance graph (at λ) is exactly equal to that induced by the connected components of the estimated concentration graph, obtained by solving the graphical lasso problem for the same λ. This characterizes a very interesting property of a path of graphical lasso solutions. Furthermore, this simple rule, when used as a wrapper around existing algorithms for the graphical lasso, leads to enormous performance gains. For a range of values of λ, our proposal splits a large graphical lasso problem into smaller tractable problems, making it possible to solve an otherwise infeasible large-scale problem. We illustrate the graceful scalability of our proposal via synthetic and real-life microarray examples.