# Regularized Estimation of Sparse Spectral Precision Matrices

arXiv: https://arxiv.org/abs/2401.11128 · HTML: https://mlsanity.com/paper/2401.11128/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2024.

## Built on by (most cited first, 4 of 4)

- [Spectral Differential Network Analysis for High-Dimensional Time Series](https://mlsanity.com/paper/2412.07905/) (2024)
- [Learning Networks from Wide-Sense Stationary Stochastic Processes](https://mlsanity.com/paper/2412.03768/) (2024)
- [A Pathwise Coordinate Descent Algorithm for LASSO Penalized Quantile Regression](https://mlsanity.com/paper/2502.12363/) (2025)
- [Inference for High-Dimensional Sparse Spectral Precision Matrices](https://mlsanity.com/paper/2606.07986/) (2026)

## Abstract (arXiv)

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.