Gaussian covariance or precision matrix estimation is a classical problem in high-dimensional data analyses. For precision matrix estimation, the graphical lasso provides an efficient approach by optimizing the log-likelihood function with $L_1$-norm penalty. Inspired by the success of graphical lasso, researchers pursue analogous outcomes for covariance matrix estimation. However, it suffers from the difficulty of non-convex optimization and a degeneration problem when $p \gt n$ due to the singularity of the sample covariance matrix. In this paper, we fix the degeneration problem by adding an extra constraint on diagonal elements. From the Bayesian perspective, a grid-point gradient descent (GPGD) algorithm together with the block Gibbs sampler is developed to sample from the posterior distribution of the correlation matrix. The algorithm provides an effective approach to draw samples under the positive-definite constraint, and can explore the whole feasible region to attain the mode of the posterior distribution. Simulation studies and a real application demonstrate that our method is competitive with other existing methods in various settings.

中文翻译：

---

高斯数据的高维相关矩阵估计：贝叶斯观点

高斯协方差或精度矩阵估计是高维数据分析中的经典问题。对于精确矩阵估计，图形套索通过以$ L_1 $-范数惩罚来优化对数似然函数，从而提供了一种有效的方法。受图形套索的成功启发，研究人员追求类似结果进行协方差矩阵估计。然而，由于样本协方差矩阵的奇异性，当$ p \ gt n $时，存在非凸优化的困难和退化问题。在本文中，我们通过在对角线元素上添加额外约束来解决退化问题。从贝叶斯角度出发，开发了网格点梯度下降（GPGD）算法和块Gibbs采样器，以从相关矩阵的后验分布中进行采样。该算法为正定约束下的样本抽取提供了一种有效的方法，并可以探索整个可行区域以获得后验分布的模式。仿真研究和实际应用表明，我们的方法在各种环境下都可以与其他现有方法竞争。