Exact Covariance Thresholding into Connected Components for Large-Scale Graphical Lasso
We consider the sparse inverse covariance regularization problem or graphical lasso with regularization parameter lambda. Suppose the sample covariance graph formed by thresholding the entries of the sample covariance matrix at lambda is decomposed into connected components. We show that the vertex-partition induced by the connected components of the thresholded sample covariance graph (at lambda) 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 lambda. 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 lambda, 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.
graphical lasso concentration graph sparse inverse covariance selection sparsity gaussian graphical models graph connected components large scale covariance estimation