  We propose an estimator for the mean of a random vector in $\mathbb{R}^d$
that can be computed in time $O(n^4+n^2d)$ for $n$ i.i.d.~samples and that has
error bounds matching the sub-Gaussian case. The only assumptions we make about
the data distribution are that it has finite mean and covariance; in
particular, we make no assumptions about higher-order moments. Like the
polynomial time estimator introduced by Hopkins, 2018, which is based on the
sum-of-squares hierarchy, our estimator achieves optimal statistical efficiency
in this challenging setting, but it has a significantly faster runtime and a
simpler analysis.
