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Sparse structure learning in high-dimensional Gaussian graphical models is animportant problem in multivariate statistical signal processing; since thesparsity pattern naturally encodes the conditional independence relationshipamong variables. However, maximum a posteriori (MAP) estimation is challengingunder hierarchical prior models, and traditional numerical optimizationroutines or expectation--maximization algorithms are difficult to implement. Tothis end, our contribution is a novel local linear approximation scheme thatcircumvents this issue using a very simple computational algorithm. Mostimportantly, the condition under which our algorithm is guaranteed to convergeto the MAP estimate is explicitly stated and is shown to cover a broad class ofcompletely monotone priors, including the graphical horseshoe. Further, theresulting MAP estimate is shown to be sparse and consistent in the$\ell_2$-norm. Numerical results validate the speed, scalability, andstatistical performance of the proposed method.

Maximum a Posteriori Estimation in Graphical Models Using Local Linear Approximation