Material for Being Robust ( in High Dimensions ) Can Be Practical

In this section, we use our filter technique to give a near sample-optimal computationally efficient algorithm to robustly estimate the mean of a sub-gaussian density with a known covariance matrix, thus proving Theorem 3.1. We emphasize that the algorithm and its analysis is essentially identical to the filtering algorithm given in Section 8.1 of [DKK+16] for the case of a Gaussian N (μ, I). The only difference is a weaker definition of the “good set of samples” (Definition 2) and a simple concentration argument (Lemma 1) showing that a random set of uncorrupted samples of the appropriate size is good with high probability. Given these, the analysis of this subsection follows straightforwardly from the analysis in Section 8.1 of [DKK+16] by plugging in the modified parameters. For the sake of completeness, we provide the details below. We start by formally defining sub-gaussian distributions: