  In this paper we initiate the study of whether or not sparse estimation tasks
can be performed efficiently in high dimensions, in the robust setting where an
$\eps$-fraction of samples are corrupted adversarially. We study the natural
robust version of two classical sparse estimation problems, namely, sparse mean
estimation and sparse PCA in the spiked covariance model. For both of these
problems, we provide the first efficient algorithms that provide non-trivial
error guarantees in the presence of noise, using only a number of samples which
is similar to the number required for these problems without noise. In
particular, our sample complexities are sublinear in the ambient dimension $d$.
Our work also suggests evidence for new computational-vs-statistical gaps for
these problems (similar to those for sparse PCA without noise) which only arise
in the presence of noise.
