Covariance Estimation: Optimal Dimension-free Guarantees for Adversarial Corruption and Heavy Tails
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We provide an estimator of the covariance matrix that achieves the optimal rate of convergence (up to constant factors) in the operator norm under two standard notions of data contamination: We allow the adversary to corrupt an η-fraction of the sample arbitrarily, while the distribution of the remaining data points only satisfies that the L_p-marginal moment with some p ≥ 4 is equivalent to the corresponding L_2-marginal moment. Despite requiring the existence of only a few moments, our estimator achieves the same tail estimates as if the underlying distribution were Gaussian. As a part of our analysis, we prove a dimension-free Bai-Yin type theorem in the regime p > 4.
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