Abstract

This paper studies the problem of accurately recovering a sparse vector from highly corrupted linear measurements is a sparse error vector whose nonzero entries may be unbounded and w is a bounded noise. We propose a so-called extended Lasso optimization which takes into consideration sparse prior information of both Our first result shows that the extended Lasso can faithfully recover both the regression and the corruption vectors. Our analysis is relied on a notion of extended restricted eigenvalue for the design matrix X. Our second set of results applies to a general class of Gaussian design matrix X with i.i.d rows for which we provide a surprising phenomenon: the extended Lasso can recover exact signed supports of both from only observations, even the fraction of corruption is arbitrarily close to one. Our analysis also shows that this amount of observations required to achieve exact signed support is optimal.