  We investigate the quantitative performance of affine-equivariant estimators
for robust mean estimation. As a natural stability requirement, the
construction of such affine-equivariant estimators has been extensively studied
in the statistics literature. We quantitatively evaluate these estimators under
two outlier models which have been the subject of much recent work: the
heavy-tailed and adversarial corruption settings. We establish lower bounds
which show that affine-equivariance induces a strict degradation in recovery
error with quantitative rates degrading by a factor of $\sqrt{d}$ in both
settings. We find that classical estimators such as the Tukey median (Tukey
'75) and Stahel-Donoho estimator (Stahel '81 and Donoho '82) are either
quantitatively sub-optimal even within the class of affine-equivariant
estimators or lack any quantitative guarantees. On the other hand, recent
estimators with strong quantitative guarantees are not affine-equivariant or
require additional distributional assumptions to achieve it. We remedy this by
constructing a new affine-equivariant estimator which nearly matches our lower
bound. Our estimator is based on a novel notion of a high-dimensional median
which may be of independent interest. Notably, our results are applicable more
broadly to any estimator whose performance is evaluated in the Mahalanobis norm
which, for affine-equivariant estimators, corresponds to an evaluation in
Euclidean norm on isotropic distributions.
