论文标题
通过惩罚对数数据启发式法的大量协方差矩阵估算
Large covariance matrix estimation via penalized log-det heuristics
论文作者
论文摘要
本文通过核标准加上$ \ ell_ {1} $ - norm norm惩罚，通过对数数据进行的大量协方差矩阵提供了一个全面的估计框架。我们开发模型框架，其中包括具有稀疏残留协方差的高维近似因素模型。我们证明，上述对数数据启发式方法是局部凸起的，具有Lipschitz连续梯度，因此可以指出近端梯度算法可以在控制阈值参数的同时数值解决问题。所提出的优化策略在一个步骤中恢复了协方差矩阵组件和潜在等级和剩余的稀疏模式，并且具有很高的概率，并且在系统上的执行不比采用Frobenius损失的相应估计量差。建立了随后的低等级和稀疏协方差矩阵估计器的误差界限，并提供了潜在几何歧管的可识别性条件，从而改善了现有文献。详尽的模拟研究和涉及欧元区银行的财务数据示例强调了概述结果的有效性。
This paper provides a comprehensive estimation framework for large covariance matrices via a log-det heuristics augmented by a nuclear norm plus $\ell_{1}$-norm penalty. We develop the model framework, which includes high-dimensional approximate factor models with a sparse residual covariance. We prove that the aforementioned log-det heuristics is locally convex with a Lipschitz-continuous gradient, so that a proximal gradient algorithm may be stated to numerically solve the problem while controlling the threshold parameters. The proposed optimization strategy recovers in a single step both the covariance matrix components and the latent rank and the residual sparsity pattern with high probability, and performs systematically not worse than the corresponding estimators employing Frobenius loss in place of the log-det heuristics. The error bounds for the ensuing low rank and sparse covariance matrix estimators are established, and the identifiability conditions for the latent geometric manifolds are provided, improving existing literature. The validity of outlined results is highlighted by an exhaustive simulation study and a financial data example involving Euro Area banks.
