  We study the algorithmic problem of estimating the mean of heavy-tailed
random vector in $\mathbb{R}^d$, given $n$ i.i.d. samples. The goal is to
design an efficient estimator that attains the optimal sub-gaussian error
bound, only assuming that the random vector has bounded mean and covariance.
Polynomial-time solutions to this problem are known but have high runtime due
to their use of semi-definite programming (SDP). Conceptually, it remains open
whether convex relaxation is truly necessary for this problem.
  In this work, we show that it is possible to go beyond SDP and achieve better
computational efficiency. In particular, we provide a spectral algorithm that
achieves the optimal statistical performance and runs in time $\widetilde
O\left(n^2 d \right)$, improving upon the previous fastest runtime $\widetilde
O\left(n^{3.5}+ n^2d\right)$ by Cherapanamjeri el al. (COLT '19). Our algorithm
is spectral in that it only requires (approximate) eigenvector computations,
which can be implemented very efficiently by, for example, power iteration or
the Lanczos method.
  At the core of our algorithm is a novel connection between the furthest
hyperplane problem introduced by Karnin et al. (COLT '12) and a structural
lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. '19).
This allows us to iteratively reduce the estimation error at a geometric rate
using only the information derived from the top singular vector of the data
matrix, leading to a significantly faster running time.
