用于稀疏精度矩阵估计的熵调整 Graphical Lasso
Entropy Adjusted Graphical Lasso for Sparse Precision Matrix Estimation
AI总结：
本文针对高维稀疏精度矩阵估计中传统收缩惩罚未显式处理不确定性的问题，提出在 Graphical Lasso 中加入对数行列式惩罚的熵调整方法，并在模拟与真实数据上验证其指标优势。
摘要：
精度矩阵估计是众多研究领域中的一个关键问题，尤其是在处理高维数据时。在此类情形下，最常见的方法是使用惩罚最大似然。相关文献通常采用 Lasso、Ridge 和 Elastic Net 范数，这些范数能够有效收缩所估计精度矩阵的元素。尽管这些收缩方法能够给出条件良好的精度矩阵估计，但它们并未显式处理与这些估计矩阵相关的不确定性。事实上，随着矩阵变得更加稀疏，精度矩阵施加的限制更少，从而导致分布具有更大的变异性，并因此产生更高的熵。本文中，我们通过加入一个额外的对数行列式惩罚项，引入了广泛使用的 Graphical Lasso 的一种熵调整扩展。所提技术的目标是对精度矩阵估计施加稀疏性，并通过对数行列式项调整不确定性。通过全面的数值分析，包括模拟数据集和真实世界数据集，评估了所提方法相较于文献中现有方法的优势。结果表明，在多项评价指标上，该方法相比文献中的现有方法具有收益。
英文摘要：
The estimation of a precision matrix is a crucial problem in various research fields, particularly when working with high dimensional data. In such settings, the most common approach is to use the penalized maximum likelihood. The literature typically employs Lasso, Ridge and Elastic Net norms, which effectively shrink the entries of the estimated precision matrix. Although these shrinkage approaches provide well-conditioned precision matrix estimates, they do not explicitly address the uncertainty associated with these estimated matrices. In fact, as the matrix becomes sparser, the precision matrix imposes fewer restrictions, leading to greater variability in the distribution, and thus, to higher entropy. In this paper, we introduce an entropy-adjusted extension of widely used Graphical Lasso using an additional log-determinant penalty term. The objective of the proposed technique is to impose sparsity on the precision matrix estimate and adjust the uncertainty through the log-determinant term. The advantage of the proposed method compared to the existing ones in the literature is evaluated through comprehensive numerical analyses, including both simulated and real-world datasets. The results demonstrate its benefits compared to existing approaches in the literature, with respect to several evaluation metrics.
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