# Large Spectral Density Matrix Estimation by Thresholding

arXiv: https://arxiv.org/abs/1812.00532 · HTML: https://mlsanity.com/paper/1812.00532/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2018.

## Built on by (most cited first, 5 of 5)

- [Learning Multi-Frequency Partial Correlation Graphs](https://mlsanity.com/paper/2311.15756/) (2023)
- [Spectral Differential Network Analysis for High-Dimensional Time Series](https://mlsanity.com/paper/2412.07905/) (2024)
- [Classification of High-dimensional Time Series in Spectral Domain using Explainable Features](https://mlsanity.com/paper/2408.08388/) (2024)
- [Learning Networks from Wide-Sense Stationary Stochastic Processes](https://mlsanity.com/paper/2412.03768/) (2024)
- [Inference for High-Dimensional Sparse Spectral Precision Matrices](https://mlsanity.com/paper/2606.07986/) (2026)

## Abstract (arXiv)

Spectral density matrix estimation of multivariate time series is a classical problem in time series and signal processing. In modern neuroscience, spectral density based metrics are commonly used for analyzing functional connectivity among brain regions. In this paper, we develop a non-asymptotic theory for regularized estimation of high-dimensional spectral density matrices of Gaussian and linear processes using thresholded versions of averaged periodograms. Our theoretical analysis ensures that consistent estimation of spectral density matrix of a p-dimensional time series using n samples is possible under high-dimensional regime log p / n rightarrow 0 as long as the true spectral density is approximately sparse. A key technical component of our analysis is a new concentration inequality of average periodogram around its expectation, which is of independent interest. Our estimation consistency results complement existing results for shrinkage based estimators of multivariate spectral density, which require no assumption on sparsity but only ensure consistent estimation in a regime p2/n rightarrow 0. In addition, our proposed thresholding based estimators perform consistent and automatic edge selection when learning coherence networks among the components of a multivariate time series. We demonstrate the advantage of our estimators using simulation studies and a real data application on functional connectivity analysis with fMRI data.