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 if the prior model admits multiple levels of hierarchy, 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 conditions under which our algorithm is guaranteed to converge to the MAP estimate are explicitly derived and are 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重要问题；因为稀疏模式自然编码了变量间的条件独立关系。然而，当先验模型允许多层次分层结构时，最大后验概率（MAP）估计具有挑战性，传统的数值优化方法或期望最大化算法难以实现。为此，我们的贡献是一种新颖的局部线性近似方案，通过非常简单的计算算法规避了这一问题。最重要的是，我们明确推导了所提算法收敛到MAP估计的条件，并证明这些条件涵盖了包括图马蹄先验在内的一大类完全单调先验。此外，所得MAP估计在$\ell_2$范数下被证明是稀疏且一致的。数值结果验证了所提方法的速度、可扩展性和统计性能。