Recovery of Sparse Signals via Conic Programming

• Activity Number: 213
• Type: Invited
• Date/Time: Monday, August 1, 2011 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Lie Wang*+ and Alexander Belloni and Victor Chernozhukov
• Companies: Massachusetts Institute of Technology and Duke University and Massachusetts Institute of Technology
• Keywords: High-dimensional sparse model ; unknown sigma ; conic programming

Abstract

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of LASSO, called square-root LASSO. The method neither relies on the knowledge of the standard deviation of the regression errors nor does it need to pre-estimate. Despite not knowing the variance, square-root LASSO achieves near-oracle performance, attaining the convergence rate that matching the performance of the standard LASSO that knows the variance. Moreover, we show that these results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions, using moderate deviation theory. Finally, we formulate the square-root LASSO as a solution to a convex conic programming problem, which allows us to use efficient computational methods, such as interior point methods, to implement the estimator.