We consider the problem of estimating sparse graphs by a lasso penalty applied to the inverse covariance matrix. Using a coordinate descent procedure for the lasso, we develop a simple algorithm— the Graphical Lasso— that is remarkably fast: it solves a 1000 node problem (∼ 500, 000 parameters) in at most a minute, and is 30 to 4000 times faster than competing methods. It also provides a concep...

Adaptive Ridge is a special form of Ridge regression, balancing the quadratic penalization on each parameter of the model. It was shown to be equivalent to Lasso (least absolute shrinkage and selection operator), in the sense that both procedures produce the same estimate. Lasso can thus be viewed as a particular quadratic penalizer. From this observation, we derive a fixed point algorithm to c...

1. A short description of the test procedure. We start by presenting the proposed test procedure in a slightly different form than in the paper. Let β̂(λ) := arg min 2‖y −Xβ‖2 + λ‖β‖1 be the Lasso estimator with tuning parameter equal to λ. The paper uses the Lasso path {β̂(λ) :λ > 0} to construct a test statistic for the significance of certain predictor variables. For a subset S ⊆ {1, . . . , p...

We devise a communication-efficient approach to distributed sparse regression in the highdimensional setting. The key idea is to average “debiased” or “desparsified” lasso estimators. We show the approach converges at the same rate as the lasso as long as the dataset is not split across too many machines, and consistently estimates the support under weaker conditions than the lasso. On the comp...

We propose a method of least squares approximation (LSA) for unified yet simple LASSO estimation. Our general theoretical framework includes ordinary least squares, generalized linear models, quantile regression, and many others as special cases. Specifically, LSA can transfer many different types of LASSO objective functions into their asymptotically equivalent least-squares problems. Thereaft...