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We consider the problem of robustifying high-dimensional structuredestimation. Robust techniques are key in real-world applications which ofteninvolve outliers and data corruption. We focus on trimmed versions ofstructurally regularized M-estimators in the high-dimensional setting,including the popular Least Trimmed Squares estimator, as well as analogousestimators for generalized linear models and graphical models, using possiblynon-convex loss functions. We present a general analysis of their statisticalconvergence rates and consistency, and then take a closer look at the trimmedversions of the Lasso and Graphical Lasso estimators as special cases. On theoptimization side, we show how to extend algorithms for M-estimators to fittrimmed variants and provide guarantees on their numerical convergence. Thegenerality and competitive performance of high-dimensional trimmed estimatorsare illustrated numerically on both simulated and real-world genomics data.

A General Family of Trimmed Estimators for Robust High-dimensional Data Analysis