Adaptive testing for the graphical lasso

We consider tests of significance in the setting of the graphical lasso for inverse covariance matrix estimation. We propose a simple test statistic based on a subsequence of the knots in the graphical lasso path. We show that this statistic has an exponential asymptotic null distribution, under the null hypothesis that the model contains the true connected components. Though the null distribution is asymptotic, we show through simulation that it provides a close approximation to the true distribution at reasonable sample sizes. Thus the test provides a simple, tractable test for the significance of new edges as they are introduced into the model. Finally, we show connections between our results and other results for regularized regression, as well as extensions of our results to other correlation matrix based methods like single-linkage clustering.