  We analyze the statistical consistency of robust estimators for precision
matrices in high dimensions. We focus on a contamination mechanism acting
cellwise on the data matrix. The estimators we analyze are formed by plugging
appropriately chosen robust covariance matrix estimators into the graphical
Lasso and CLIME. Such estimators were recently proposed in the robust
statistics literature, but only analyzed mathematically from the point of view
of the breakdown point. This paper provides complementary high-dimensional
error bounds for the precision matrix estimators that reveal the interplay
between the dimensionality of the problem and the degree of contamination
permitted in the observed distribution. We also show that although the
graphical Lasso and CLIME estimators perform equally well from the point of
view of statistical consistency, the breakdown property of the graphical Lasso
is superior to that of CLIME. We discuss implications of our work for problems
involving graphical model estimation when the uncontaminated data follow a
multivariate normal distribution, and the goal is to estimate the support of
the population-level precision matrix. Our error bounds do not make any
assumptions about the the contaminating distribution and allow for a
nonvanishing fraction of cellwise contamination.
