We present a simple and effective algorithm for the problem of sparse robust linear regression. In this problem, one would like to estimate a sparse vector w * ∈ R n from linear measurements corrupted by sparse noise that can arbitrarily change an adversarially chosen η fraction of measured responses y, as well as introduce bounded norm noise to the responses. For Gaussian measurements, we show that a simple algorithm based on L1 regression can successfully estimate w * for any η < η 0 ≈ 0.239, and that this threshold is tight for the algorithm. The number of measurements required by the algorithm is O(k log n k) for k-sparse estimation, which is within constant factors of the number needed without any sparse noise. Of the three properties we show-the ability to estimate sparse, as well as dense, w * ; the tolerance of a large constant fraction of outliers; and tolerance of adversarial rather than distri-butional (e.g., Gaussian) dense noise-to the best of our knowledge, no previous result achieved more than two.
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