  This paper studies the problem of accurately recovering a sparse vector
$\beta^{\star}$ from highly corrupted linear measurements $y = X \beta^{\star}
+ e^{\star} + w$ where $e^{\star}$ 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 $\beta^{\star}$ and $e^{\star}$. Our first result shows
that the extended Lasso can faithfully recover both the regression as well as
the corruption vector. Our analysis relies on the 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 $\oper N(0,
\Sigma)$, for which we can establish a surprising result: the extended Lasso
can recover exact signed supports of both $\beta^{\star}$ and $e^{\star}$ from
only $\Omega(k \log p \log n)$ observations, even when 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 indeed
optimal.
