  We study the problem of testing the covariance matrix of a high-dimensional
Gaussian in a robust setting, where the input distribution has been corrupted
in Huber's contamination model. Specifically, we are given i.i.d. samples from
a distribution of the form $Z = (1-\epsilon) X + \epsilon B$, where $X$ is a
zero-mean and unknown covariance Gaussian $\mathcal{N}(0, \Sigma)$, $B$ is a
fixed but unknown noise distribution, and $\epsilon>0$ is an arbitrarily small
constant representing the proportion of contamination. We want to distinguish
between the cases that $\Sigma$ is the identity matrix versus $\gamma$-far from
the identity in Frobenius norm.
  In the absence of contamination, prior work gave a simple tester for this
hypothesis testing task that uses $O(d)$ samples. Moreover, this sample upper
bound was shown to be best possible, within constant factors. Our main result
is that the sample complexity of covariance testing dramatically increases in
the contaminated setting. In particular, we prove a sample complexity lower
bound of $\Omega(d^2)$ for $\epsilon$ an arbitrarily small constant and $\gamma
= 1/2$. This lower bound is best possible, as $O(d^2)$ samples suffice to even
robustly {\em learn} the covariance. The conceptual implication of our result
is that, for the natural setting we consider, robust hypothesis testing is at
least as hard as robust estimation.
