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.

本文研究从高度破损的线性测量中准确恢复稀疏向量 β* 的问题，提出了考虑 β* 和稀疏误差向量 e * 的稀疏先验信息的扩展 Lasso 优化方法，该方法可以从只有 Ω(klogplogn) 次观测中精确地恢复 β* 和 e * 的精确符号支持。