This paper provides a comprehensive estimation framework for large covariance matrices via a log-det heuristics augmented by a nuclear norm plus $l_{1}$ norm penalty. %We develop the model framework, which includes high-dimensional approximate factor models with a sparse residual covariance. The underlying assumptions allow for non-pervasive latent eigenvalues and a prominent residual covariance pattern. 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 with high probability both the covariance matrix components and the latent rank and the residual sparsity pattern, 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 condition for the latent geometric manifolds is provided. The validity of outlined results is highlighted by means of an exhaustive simulation study and a real financial data example involving euro zone banks.
翻译:本文为大共变矩阵提供了一个全面的估算框架,通过对数-偏差法,辅之以核规范加上$l ⁇ 1美元的标准罚款。% 我们开发了模型框架,其中包括高维近效系数模型,其残余的共差少。 基本假设允许非渗透潜潜潜潜潜潜偏值和显著余余余共差模式。 我们证明,上述对数- 偏差变异法与Lipschitz持续梯度的偏差是局部的,因此可以说明一种准氧化梯度算法,以便在控制临界参数时从数字上解决问题。 拟议的优化战略以高概率恢复共变矩阵组成部分和潜在等级以及残余聚度模式,并系统地进行不比相应的估计者更差的工作。 将随后的低级和稀疏差的易变差矩阵估计者设定出错误界限,并将潜在几何矩阵参数的识别性条件用于控制阈值参数。 所提出的优化战略以极可能的方式回收共变基矩阵组件和残余体结构模式,并系统地进行精确的模拟,从而突出地展示了对欧元进行彻底研究的结果。