Equivalence of graphical lasso and thresholding for sparse graphs

This paper is concerned with the problem of finding a sparse graph capturing the conditional dependence between the entries of a Gaussian random vector, where the only available information is a sample correlation matrix. A popular approach is to solve a graphical lasso problem with a sparsity-promoting regularization term. This paper derives a simple condition under which the computationally-expensive graphical lasso behaves the same as the simple heuristic method of thresholding. This condition depends only on the solution of graphical lasso and makes no direct use of the sample correlation matrix or the regularization coefficient. It is also proved that this condition is always satisfied if the solution of graphical lasso is replaced by its first-order Taylor approximation. The condition is tested on several random problems and it is shown that graphical lasso and the thresholding method (based on the correlation matrix) lead to a similar result (if not equivalent), provided the regularization term is high enough to seek a sparse graph.