Robust sub-Gaussian estimation of a mean vector in nearly linear time

We construct an algorithm for estimating the mean of a heavy-tailed random variable when given adversarial corrupted sample N independent observations. The only assumption we make on distribution noncorrupted (or informative) data is existence covariance matrix Σ, unknown to statistician. Our outputs μˆ, which robust presence |O| outliers and satisfies (1)‖μˆ−μ‖2≲ Tr(Σ) N+ ‖Σ‖opK with probability at least 1−exp(−c0K)−exp(−c1u), runtime O˜(Nd+uKd) where K∈{600|O|,…,N} u∈N∗ are two parameters algorithm. fully data-dependent does not use (1) in its construction, combines recently developed tools median-of-means estimators covering semidefinite programming. also show that this can automatically adapt number (adaptive choice K) it same bound expectation.