  We propose an l1-regularized likelihood method for estimating the inverse
covariance matrix in the high-dimensional multivariate normal model in presence
of missing data. Our method is based on the assumption that the data are
missing at random (MAR) which entails also the completely missing at random
case. The implementation of the method is non-trivial as the observed negative
log-likelihood generally is a complicated and non-convex function. We propose
an efficient EM algorithm for optimization with provable numerical convergence
properties. Furthermore, we extend the methodology to handle missing values in
a sparse regression context. We demonstrate both methods on simulated and real
data.
