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.

翻译：高维高斯图模型中的稀疏结构学习是多变量统计信号处理中的一个重要问题，因为稀疏模式天然地编码了变量间的条件独立关系。然而，在分层先验模型下，最大后验估计面临挑战，传统的数值优化方法或期望最大化算法难以实现。为此，我们的贡献在于提出了一种新颖的局部线性近似方案，该方案通过一种非常简单的计算算法规避了这一问题。最重要的是，我们明确给出了算法保证收敛到最大后验估计的条件，并证明该条件涵盖了包括graphical horseshoe在内的广泛完全单调先验类。此外，所得最大后验估计被证明是稀疏的且在$\ell_2$范数下具有一致性。数值结果验证了所提方法的速度、可扩展性和统计性能。