  Estimating conditional dependence graphs and precision matrices are some of
the most common problems in modern statistics and machine learning. When data
are fully observed, penalized maximum likelihood-type estimators have become
standard tools for estimating graphical models under sparsity conditions.
Extensions of these methods to more complex settings where data are
contaminated with additive or multiplicative noise have been developed in
recent years. In these settings, however, the relative performance of different
methods is not well understood and algorithmic gaps still exist. In particular,
in high-dimensional settings these methods require using non-positive
semidefinite matrices as inputs, presenting novel optimization challenges. We
develop an alternating direction method of multipliers (ADMM) algorithm for
these problems, providing a feasible algorithm to estimate precision matrices
with indefinite input and potentially nonconvex penalties. We compare this
method with existing alternative solutions and empirically characterize the
tradeoffs between them. Finally, we use this method to explore the networks
among US senators estimated from voting records data.
