  We study the problem of high-dimensional covariance estimation under the
constraint that the partial correlations are nonnegative. The sign constraints
dramatically simplify estimation: the Gaussian maximum likelihood estimator is
well defined with only two observations regardless of the number of variables.
We analyze its performance in the setting where the dimension may be much
larger than the sample size. We establish that the estimator is both
high-dimensionally consistent and minimax optimal in the symmetrized Stein
loss. We also prove a negative result which shows that the sign-constraints can
introduce substantial bias for estimating the top eigenvalue of the covariance
matrix.
