  We study the robust mean estimation problem in high dimensions, where $\alpha
<0.5$ fraction of the data points can be arbitrarily corrupted. Motivated by
compressive sensing, we formulate the robust mean estimation problem as the
minimization of the $\ell_0$-`norm' of the outlier indicator vector, under
second moment constraints on the inlier data points. We prove that the global
minimum of this objective is order optimal for the robust mean estimation
problem, and we propose a general framework for minimizing the objective. We
further leverage the $\ell_1$ and $\ell_p$ $(0<p<1)$, minimization techniques
in compressive sensing to provide computationally tractable solutions to the
$\ell_0$ minimization problem. Both synthetic and real data experiments
demonstrate that the proposed algorithms significantly outperform
state-of-the-art robust mean estimation methods.
