  We present a simple and effective algorithm for the problem of \emph{sparse
robust linear regression}. In this problem, one would like to estimate a sparse
vector $w^* \in \mathbb{R}^n$ from linear measurements corrupted by sparse
noise that can arbitrarily change an adversarially chosen $\eta$ fraction of
measured responses $y$, as well as introduce bounded norm noise to the
responses. For Gaussian measurements, we show that a simple algorithm based on
L1 regression can successfully estimate $w^*$ for any $\eta < \eta_0 \approx
0.239$, and that this threshold is tight for the algorithm. The number of
measurements required by the algorithm is $O(k \log \frac{n}{k})$ for
$k$-sparse estimation, which is within constant factors of the number needed
without any sparse noise. Of the three properties we show---the ability to
estimate sparse, as well as dense, $w^*$; the tolerance of a large constant
fraction of outliers; and tolerance of adversarial rather than distributional
(e.g., Gaussian) dense noise---to the best of our knowledge, no previous result
achieved more than two.
