  We study the problem of high-dimensional sparse mean estimation in the
presence of an $\epsilon$-fraction of adversarial outliers. Prior work obtained
sample and computationally efficient algorithms for this task for
identity-covariance subgaussian distributions. In this work, we develop the
first efficient algorithms for robust sparse mean estimation without a priori
knowledge of the covariance. For distributions on $\mathbb R^d$ with
"certifiably bounded" $t$-th moments and sufficiently light tails, our
algorithm achieves error of $O(\epsilon^{1-1/t})$ with sample complexity $m =
(k\log(d))^{O(t)}/\epsilon^{2-2/t}$. For the special case of the Gaussian
distribution, our algorithm achieves near-optimal error of $\tilde O(\epsilon)$
with sample complexity $m = O(k^4 \mathrm{polylog}(d))/\epsilon^2$. Our
algorithms follow the Sum-of-Squares based, proofs to algorithms approach. We
complement our upper bounds with Statistical Query and low-degree polynomial
testing lower bounds, providing evidence that the sample-time-error tradeoffs
achieved by our algorithms are qualitatively the best possible.
