Abstract: In high dimensions, 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 such high-dimensional applications 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. Data and simulations indicate that our method can be useful in improving variable selection performances.