  We study the fundamental problem of learning the parameters of a
high-dimensional Gaussian in the presence of noise -- where an
$\varepsilon$-fraction of our samples were chosen by an adversary. We give
robust estimators that achieve estimation error $O(\varepsilon)$ in the total
variation distance, which is optimal up to a universal constant that is
independent of the dimension.
  In the case where just the mean is unknown, our robustness guarantee is
optimal up to a factor of $\sqrt{2}$ and the running time is polynomial in $d$
and $1/\epsilon$. When both the mean and covariance are unknown, the running
time is polynomial in $d$ and quasipolynomial in $1/\varepsilon$. Moreover all
of our algorithms require only a polynomial number of samples. Our work shows
that the same sorts of error guarantees that were established over fifty years
ago in the one-dimensional setting can also be achieved by efficient algorithms
in high-dimensional settings.
