  We explore the connection between outlier-robust high-dimensional statistics
and non-convex optimization in the presence of sparsity constraints, with a
focus on the fundamental tasks of robust sparse mean estimation and robust
sparse PCA. We develop novel and simple optimization formulations for these
problems such that any approximate stationary point of the associated
optimization problem yields a near-optimal solution for the underlying robust
estimation task. As a corollary, we obtain that any first-order method that
efficiently converges to stationarity yields an efficient algorithm for these
tasks. The obtained algorithms are simple, practical, and succeed under broader
distributional assumptions compared to prior work.
