  We consider the sparse inverse covariance regularization problem or graphical
lasso with regularization parameter $\rho$. Suppose the co- variance graph
formed by thresholding the entries of the sample covariance matrix at $\rho$ is
decomposed into connected components. We show that the vertex-partition induced
by the thresholded covariance graph is exactly equal to that induced by the
estimated concentration graph. This simple rule, when used as a wrapper around
existing algorithms, leads to enormous performance gains. For large values of
$\rho$, our proposal splits a large graphical lasso problem into smaller
tractable problems, making it possible to solve an otherwise infeasible large
scale graphical lasso problem.
