  Sparse structure learning in high-dimensional Gaussian graphical models is an
important problem in multivariate statistical signal processing; since the
sparsity pattern naturally encodes the conditional independence relationship
among variables. However, maximum a posteriori (MAP) estimation is challenging
under hierarchical prior models, and traditional numerical optimization
routines or expectation--maximization algorithms are difficult to implement. To
this end, our contribution is a novel local linear approximation scheme that
circumvents this issue using a very simple computational algorithm. Most
importantly, the condition under which our algorithm is guaranteed to converge
to the MAP estimate is explicitly stated and is shown to cover a broad class of
completely monotone priors, including the graphical horseshoe. Further, the
resulting MAP estimate is shown to be sparse and consistent in the
$\ell_2$-norm. Numerical results validate the speed, scalability, and
statistical performance of the proposed method.

Methodology;Computation