论文标题
在总阳性下对图形模型的自适应估计
Adaptive Estimation of Graphical Models under Total Positivity
论文作者
论文摘要
我们将估计（对角线占主导地位）M-矩阵的问题视为高斯图形模型中的精确矩阵。这些模型表现出有趣的特性，例如，最大似然估计量的存在只有两个观察到M-Matrices \ citep {Lauritzen2019maximim，Slawski2015Simimation}，甚至是对角度优势M-Matrices \ citep \ citep {truellp {truellyper {truellaximien}}的凝结。我们提出了一种自适应的多阶段估计方法，该方法通过在每个阶段解决加权$ \ ell_1 $ regarlized问题来完善估计值。此外，我们基于梯度投影方法开发了一个统一的框架来解决正规化问题，并结合了不同的投影来处理M-矩阵的约束和对角线主导的M-矩阵。提供了对估计误差的理论分析。我们的方法在精确矩阵估计和图形边缘识别方面优于最先进的方法，如合成和财务时间序列数据集所证明的那样。
We consider the problem of estimating (diagonally dominant) M-matrices as precision matrices in Gaussian graphical models. These models exhibit intriguing properties, such as the existence of the maximum likelihood estimator with merely two observations for M-matrices \citep{lauritzen2019maximum,slawski2015estimation} and even one observation for diagonally dominant M-matrices \citep{truell2021maximum}. We propose an adaptive multiple-stage estimation method that refines the estimate by solving a weighted $\ell_1$-regularized problem at each stage. Furthermore, we develop a unified framework based on the gradient projection method to solve the regularized problem, incorporating distinct projections to handle the constraints of M-matrices and diagonally dominant M-matrices. A theoretical analysis of the estimation error is provided. Our method outperforms state-of-the-art methods in precision matrix estimation and graph edge identification, as evidenced by synthetic and financial time-series data sets.
