  This paper confirms a surprising phenomenon first observed by Wright
\textit{et al.} \cite{WYGSM_Face_2009_J} \cite{WM_denseError_2010_J} under
different setting: given $m$ highly corrupted measurements $y = A_{\Omega
\bullet} x^{\star} + e^{\star}$, where $A_{\Omega \bullet}$ is a submatrix
whose rows are selected uniformly at random from rows of an orthogonal matrix
$A$ and $e^{\star}$ is an unknown sparse error vector whose nonzero entries may
be unbounded, we show that with high probability $\ell_1$-minimization can
recover the sparse signal of interest $x^{\star}$ exactly from only $m = C
\mu^2 k (\log n)^2$ where $k$ is the number of nonzero components of
$x^{\star}$ and $\mu = n \max_{ij} A_{ij}^2$, even if nearly 100% of the
measurements are corrupted. We further guarantee that stable recovery is
possible when measurements are polluted by both gross sparse and small dense
errors: $y = A_{\Omega \bullet} x^{\star} + e^{\star}+ \nu$ where $\nu$ is the
small dense noise with bounded energy. Numerous simulation results under
various settings are also presented to verify the validity of the theory as
well as to illustrate the promising potential of the proposed framework.
