Robust Mean Estimation in High Dimensions via ℓ_0 Minimization
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We study the robust mean estimation problem in high dimensions, where α <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 ℓ_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 ℓ_1 and ℓ_p (0<p<1), minimization techniques in compressive sensing to provide computationally tractable solutions to the ℓ_0 minimization problem. Both synthetic and real data experiments demonstrate that the proposed algorithms significantly outperform state-of-the-art robust mean estimation methods.
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