for K linear regressions. The support union of K p-dimensional regression vectors (collected as columns of matrix B∗) is recovered using l1/l2-regularized Lasso. Sufficient and necessary conditions on sample complexity are characterized as a sharp threshold to guarantee successful recovery of the support union. This model has been previously studied via l1/l∞regularized Lasso by Negahban & Wain...

Binary logistic regression with a sparsity constraint on the solution plays a vital role in many high dimensional machine learning applications. In some cases, the features can be grouped together, so that entire subsets of features can be selected or zeroed out. In many applications, however, this can be very restrictive. In this paper, we are interested in a less restrictive form of structure...

Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weighted L1-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by the L1-penalty. In the ul...

I use the adaptive elastic net in a Bayesian framework and test its forecasting performance against lasso, adaptive lasso and elastic net (all used in a Bayesian framework) in a series of simulations, as well as in an empirical exercise for macroeconomic Euro area data. The results suggest that elastic net is the best model among the four Bayesian methods considered. Adaptive lasso, on the othe...

LASSO is a useful method for achieving both shrinkage and variable selection simultaneously. The main idea of LASSO is to use the L1 constraint in the regularization step which has been applied to various models such as wavelets, kernel machines, smoothing splines, and multiclass logistic models. We call such models with the L1 constraint generalized LASSO models. In this paper, we propose a ne...

This paper develops a theory for group Lasso using a concept called strong group sparsity. Our result shows that group Lasso is superior to standard Lasso for strongly group-sparse signals. This provides a convincing theoretical justi cation for using group sparse regularization when the underlying group structure is consistent with the data. Moreover, the theory predicts some limitations of th...

Group lasso is a natural extension of lasso and selects variables in a grouped manner. However, group lasso suffers from estimation inefficiency and selection inconsistency. To remedy these problems, we propose the adaptive group lasso method. We show theoretically that the new method is able to identify the true model consistently, and the resulting estimator can be as efficient as oracle. Num...

The Lasso is an attractive approach to variable selection in sparse, highdimensional regression models. Much work has been done to study the selection and estimation properties of the Lasso in the context of least squares regression. However, the least squares based method is sensitive to outliers. An alternative to the least squares method is the least absolute deviations (LAD) method which is...

We study the validity of the pairs bootstrap for Lasso estimators in linear regression models with random covariates and heteroscedastic error terms. We show that the naive pairs bootstrap does not consistently estimate the distribution of the Lasso estimator. In particular, we identify two different sources for the failure of the bootstrap. First, in the bootstrap samples the Lasso estimator f...