Much theoretical and applied work has been devoted to high-dimensional regression with clean data. However, we often face corrupted data in many applications where missing data and measurement errors cannot be ignored. Loh and Wainwright [Ann. Statist. 40 (2012) 1637–1664] proposed a nonconvex modification of the Lasso for doing high-dimensional regression with noisy and missing data. It is generally agreed that the virtues of convexity contribute fundamentally the success and popularity of the Lasso. In light of this, we propose a new method named CoCoLasso that is convex and can handle a general class of corrupted datasets. We establish the estimation error bounds of CoCoLasso and its asymptotic sign-consistent selection property. We further elucidate how the standard cross validation techniques can be misleading in presence of measurement error and develop a novel calibrated cross-validation technique by using the basic idea in CoCoLasso. The calibrated cross-validation has its own importance. We demonstrate the superior performance of our method over the nonconvex approach by simulation studies.
Bibliographical noteFunding Information:
Received November 2015; revised August 2016. 1Supported in part by NSF Grant DMS-15-05111. MSC2010 subject classifications. Primary 62J07; secondary 62F12. Key words and phrases. Convex optimization, error in variables, high-dimensional LASSO, missing data.
© Institute of Mathematical Statistics, 2017.
- Convex optimization
- Error in variables
- High-dimensional regression
- Missing data