The Sample Complexity of Robust Covariance Testing
Abstract
We study the problem of testing the covariance matrix of a highdimensional 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 zeromean 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.15802
 Bibcode:
 2020arXiv201215802D
 Keywords:

 Computer Science  Machine Learning;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning