  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 (2012) proposed a non-convex 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
including the cases of additive measurement error and random missing data. 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 corrected cross-validation technique by using the basic idea in
CoCoLasso. The corrected cross-validation has its own importance. We
demonstrate the superior performance of our method over the non-convex approach
by simulation studies.
