We extend the `2-consistency result of (Meinshausen and Yu 2008) from the Lasso to the group Lasso. Our main theorem shows that the group Lasso achieves estimation consistency under a mild condition and an asymptotic upper bound on the number of selected variables can be obtained. As a result, we can apply the nonnegative garrote procedure to the group Lasso result to obtain an estimator which ...

The LASSO (Tibshirani, J R Stat Soc Ser B 58(1):267–288, 1996, [30]) and the adaptive LASSO (Zou, J Am Stat Assoc 101:1418–1429, 2006, [37]) are popular in regression analysis for their advantage of simultaneous variable selection and parameter estimation, and also have been applied to autoregressive time series models. We propose the doubly adaptive LASSO (daLASSO), or PLAC-weighted adaptive L...

This paper presents an upper bound for the estimation error of the constrained lasso, under the high-dimensional (n < p) setting. In contrast to existing results, the error bound in this paper is sharp, is valid when the parameter to be estimated is not exactly sparse (e.g., when the parameter is weakly sparse), and shows explicitly the effect of over-estimating the `1-norm of the parameter to ...

The literature is replete with variable selection techniques for the classical linear regression model. It is only relatively recently that authors have begun to explore variable selection in fully nonparametric and additive regression models. One such variable selection technique is a generalization of the LASSO called the group LASSO. In this work, we demonstrate a connection between the grou...

During an electron-microscopic survey with the aim of identifying the parvovirus MVM transcription template, we observed previously unidentified structures of MVM DNA in lysates of virus-infected cells. These included double-stranded "lasso"-like structures and relaxed circles. Both structures were of unit length MVM DNA, indicating that they were not intermediates formed during replication; th...

In this paper, we are concerned with regression problems where covariates can be grouped in nonoverlapping blocks, and where only a few of them are assumed to be active. In such a situation, the group Lasso is an attractive method for variable selection since it promotes sparsity of the groups. We study the sensitivity of any group Lasso solution to the observations and provide its precise loca...

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...