Estimation of a sparse spectral precision matrix, the inverse of a spectral density matrix, is a canonical problem in frequency-domain analysis of high-dimensional time series (HDTS), with applications in neurosciences and environmental sciences. Existing estimators use off-the-shelf optimizers for complex variables that limit scalability, uniform (non-adaptive) penalization that is not tailored to handle heterogeneity across time series components, and lack a formal non-asymptotic theory that systematically analyzes approximation and estimation errors in high-dimension. In this work, develop fast pathwise coordinate descent (CD) algorithms and non-asymptotic theory for a complex graphical lasso (CGLASSO) and an adaptive version CAGLASSO, that adapts penalization to the underlying scale of variability. For fast algorithms, we devise a realification procedure based on ring isomorphism, a notion from abstract algebra, that can be used for other high-dimensional optimization problems over complex variables. Our non-asymptotic analysis shows that consistency is possible in high-dimension under suitable sparsity assumptions. A key step is to separately bound the approximation and estimation error arising from treating the finite-sample discrete Fourier Transforms (DFTs) as i.i.d. complex-valued data, an issue well-addressed in classical time series but relatively less explored in HDTS literature. We demonstrate the performance of our proposed estimators in several simulated data sets and a real data application from neuroscience.

翻译：稀疏谱精度矩阵（即谱密度矩阵的逆）的估计是高维时间序列频域分析中的一个经典问题，在神经科学和环境科学等领域具有重要应用。现有估计方法通常采用针对复变量的现成优化器，其可扩展性受限；使用均匀（非自适应）惩罚项，未能针对时间序列分量间的异质性进行定制；且缺乏系统的非渐近理论来严格分析高维情形下的近似误差与估计误差。本研究针对复图套索及其自适应版本（分别记为CGLASSO与CAGLASSO）提出了快速路径坐标下降算法及非渐近理论，其中自适应版本能根据变异性尺度调整惩罚强度。为实现快速算法，我们基于抽象代数中的环同构概念设计了一种实化转换方法，该方法可推广至其他涉及复变量的高维优化问题。非渐近分析表明，在适当的稀疏性假设下，高维情形中仍可保证估计的一致性。关键步骤在于分别界定由有限样本离散傅里叶变换被视为独立同分布复值数据所产生的近似误差与估计误差——该问题在经典时间序列分析中已有充分探讨，但在高维时间序列文献中研究相对较少。我们通过多组模拟数据及一项神经科学实际数据应用验证了所提估计方法的性能。