  This paper addresses the task of estimating a covariance matrix under a
patternless sparsity assumption. In contrast to existing approaches based on
thresholding or shrinkage penalties, we propose a likelihood-based method that
regularizes the distance from the covariance estimate to a symmetric sparsity
set. This formulation avoids unwanted shrinkage induced by more common norm
penalties and enables optimization of the resulting non-convex objective by
solving a sequence of smooth, unconstrained subproblems. These subproblems are
generated and solved via the proximal distance version of the
majorization-minimization principle. The resulting algorithm executes rapidly,
gracefully handles settings where the number of parameters exceeds the number
of cases, yields a positive definite solution, and enjoys desirable convergence
properties. Empirically, we demonstrate that our approach outperforms competing
methods by several metrics across a suite of simulated experiments. Its merits
are illustrated on an international migration dataset and a classic case study
on flow cytometry. Our findings suggest that the marginal and conditional
dependency networks for the cell signalling data are more similar than
previously concluded.
