两种用于稀疏协方差矩阵最大似然估计的新算法及其在图建模中的应用
Two new algorithms for maximum likelihood estimation of sparse covariance matrices with applications to graphical modeling
AI总结：
本文提出两种基于协方差图的两阶段稀疏协方差矩阵最大似然估计算法，分别采用块坐标下降与近端距离法，在样本数低于维度时仍有效，且估计精度优于两种现有方法。
摘要：
在本文中，我们提出了两种用于高维稀疏协方差矩阵最大似然估计（MLE）的新算法。与大多数现有最先进方法不同，这些方法要么使用正则化技术，要么对似然函数施加惩罚以强制稀疏性，我们则基于估计的协方差图来解决MLE问题。更具体地说，我们提出了一种两阶段程序：在第一阶段，我们使用错误发现率（FDR）的多重假设检验方法来确定目标协方差矩阵的稀疏模式（换言之，在高斯图模型下协方差图中的边际独立性）；在第二阶段，我们使用块坐标下降方法来估计非零值，或使用近端距离方法，该方法对估计的协方差图与目标协方差矩阵之间的距离进行惩罚。这样做产生了两种不同的方法，每种方法各有其优势：坐标下降方法不需要调整任何超参数，而近端距离方法计算速度快，但需要仔细调整惩罚参数。两种方法即使在观测样本数少于数据维度的情况下也有效。为了评估性能，我们在模拟数据和真实数据上测试了所提出的方法，并表明它们比两种最先进的方法提供了更准确的稀疏协方差矩阵估计。
英文摘要：
In this paper, we propose two new algorithms for maximum-likelihood estimation (MLE) of high dimensional sparse covariance matrices. Unlike most of the state of-the-art methods, which either use regularization techniques or penalize the likelihood to impose sparsity, we solve the MLE problem based on an estimated covariance graph. More specifically, we propose a two-stage procedure: in the first stage, we determine the sparsity pattern of the target covariance matrix (in other words the marginal independence in the covariance graph under a Gaussian graphical model) using the multiple hypothesis testing method of false discovery rate (FDR), and in the second stage we use either a block coordinate descent approach to estimate the non-zero values or a proximal distance approach that penalizes the distance between the estimated covariance graph and the target covariance matrix. Doing so gives rise to two different methods, each with its own advantage: the coordinate descent approach does not require tuning of any hyper-parameters, whereas the proximal distance approach is computationally fast but requires a careful tuning of the penalty parameter. Both methods are effective even in cases where the number of observed samples is less than the dimension of the data. For performance evaluation, we test the proposed methods on both simulated and real-world data and show that they provide more accurate estimates of the sparse covariance matrix than two state-of-the-art methods.
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