  Robust estimation is much more challenging in high dimensions than it is in
one dimension: Most techniques either lead to intractable optimization problems
or estimators that can tolerate only a tiny fraction of errors. Recent work in
theoretical computer science has shown that, in appropriate distributional
models, it is possible to robustly estimate the mean and covariance with
polynomial time algorithms that can tolerate a constant fraction of
corruptions, independent of the dimension. However, the sample and time
complexity of these algorithms is prohibitively large for high-dimensional
applications. In this work, we address both of these issues by establishing
sample complexity bounds that are optimal, up to logarithmic factors, as well
as giving various refinements that allow the algorithms to tolerate a much
larger fraction of corruptions. Finally, we show on both synthetic and real
data that our algorithms have state-of-the-art performance and suddenly make
high-dimensional robust estimation a realistic possibility.
