  In high-dimensions, many variable selection methods, such as the lasso, are
often limited by excessive variability and rank deficiency of the sample
covariance matrix. Covariance sparsity is a natural phenomenon in
high-dimensional applications, such as microarray analysis, image processing,
etc., in which a large number of predictors are independent or weakly
correlated. In this paper, we propose the covariance-thresholded lasso, a new
class of regression methods that can utilize covariance sparsity to improve
variable selection. We establish theoretical results, under the random design
setting, that relate covariance sparsity to variable selection. Real-data and
simulation examples indicate that our method can be useful in improving
variable selection performances.
