  We consider the problem of robustifying high-dimensional structured
estimation. Robust techniques are key in real-world applications which often
involve outliers and data corruption. We focus on trimmed versions of
structurally regularized M-estimators in the high-dimensional setting,
including the popular Least Trimmed Squares estimator, as well as analogous
estimators for generalized linear models and graphical models, using possibly
non-convex loss functions. We present a general analysis of their statistical
convergence rates and consistency, and then take a closer look at the trimmed
versions of the Lasso and Graphical Lasso estimators as special cases. On the
optimization side, we show how to extend algorithms for M-estimators to fit
trimmed variants and provide guarantees on their numerical convergence. The
generality and competitive performance of high-dimensional trimmed estimators
are illustrated numerically on both simulated and real-world genomics data.
