Inference for High-Dimensional Sparse Spectral Precision Matrices

Technical Summary: Inference for High-Dimensional Sparse Spectral Precision Matrices

Problem Statement
Gaussian graphical models (GGMs) in the spectral domain offer a principled approach for recovering conditional dependence structures in stationary high-dimensional time series. The Spectral Precision Matrix (SPM), $\Theta(\omega) = f(\omega)^{-1}$, at a fixed frequency $\omega$ encodes the absence or presence of conditional relationships among time series components. While existing literature has established methods for estimating SPMs using regularized optimization (e.g., Complex Graphical Lasso) and for low-dimensional inference on spectral density, a rigorous framework for entry-wise statistical inference (confidence intervals and hypothesis tests) in high-dimensional settings is lacking.

The problem is technically challenging due to three primary factors:

1. Complex-valued nature: The SPM and its estimators are complex-valued, complicating variance estimation compared to real-valued i.i.d. settings.
2. Temporal dependence: Finite-sample Discrete Fourier Transforms (DFTs) induce truncation and smoothing biases, and DFTs at nearby frequencies are not strictly independent.
3. High-dimensional bias: Standard debiasing techniques for i.i.d. data (e.g., Jankova et al., 2017) do not directly apply because they fail to account for the specific approximation errors introduced by spectral smoothing and the finite-sample covariance structure of DFTs. Existing approaches, such as nodewise regression on DFTs, do not fully exploit the likelihood structure of neighboring frequencies.

Methodology
The authors propose a high-dimensional inference framework centered on a debiased complex graphical lasso (deCGLASSO) estimator.

1. Estimator Construction:

   • The method begins with the Complex Graphical Lasso (CGLASSO) estimator, $\hat{\Theta}_j$, which minimizes an approximate full likelihood of neighboring DFTs at frequency $\omega_j$.
   • To correct the regularization bias inherent in the Lasso, the authors adapt the debiasing technique of Jankova et al. (2017) to the complex spectral domain. The debiased estimator is defined as:
     $$\tilde{\Theta}_j = \hat{\Theta}_j + \lambda \hat{\Theta}_j \hat{\Psi} \hat{\Theta}_j = 2\hat{\Theta}_j - \hat{\Theta}_j \hat{f}_j \hat{\Theta}_j$$
     where $\hat{f}_j$ is the averaged periodogram estimator and $\hat{\Psi}$ is the sub-gradient of the penalty term. This construction retains the likelihood-based structure of CGLASSO, utilizing information from neighboring Fourier frequencies.

2. Asymptotic Theory:

   • The authors establish the asymptotic normality of the real and imaginary parts of the debiased estimator entries.
   • A key theoretical step involves extending Central Limit Theorem (CLT) results for quadratic forms of stationary time series (Wu and Zaffaroni, 2014) to a joint CLT for collections of complex-valued quadratic forms.
   • The authors rigorously control the regularization bias, finite-sample truncation bias, and smoothing bias. They show that under suitable regularity conditions (including sparsity and incoherence assumptions), the bias terms vanish asymptotically, leaving the leading term determined by the quadratic form of the DFTs.

3. Variance Estimation and Inference:

   • To construct entry-wise confidence regions, the authors derive the asymptotic covariance structure for the real and imaginary parts of the estimator.
   • They propose two consistent estimators for this covariance:
     • A plug-in estimator based on population-level variance and pseudovariance expressions.
     • A Heteroscedasticity and Autocorrelation Consistent (HAC) estimator that accounts for dependence across nearby DFTs by weighting pairs of DFTs.
   • Using these estimators, they construct joint confidence ellipsoids for the complex entries of the SPM and develop a multiple testing procedure to control the False Discovery Rate (FDR) across all off-diagonal entries.

Key Contributions

1. Debiased Likelihood-Based Estimator: Introduction of the deCGLASSO estimator, which adapts debiasing to the complex spectral domain while leveraging the full likelihood of neighboring DFTs, offering potential advantages over nodewise regression approaches.
2. Theoretical Guarantees: Establishment of entry-wise asymptotic normality for the debiased SPM estimator. This includes a novel proof technique that simultaneously controls regularization, truncation, and smoothing biases, extending existing CLT results to complex-valued multivariate time series.
3. Feasible Inference Framework: Development of consistent variance estimators (plug-in and HAC) and a formal procedure for constructing confidence intervals and controlling FDR for frequency-specific conditional associations.

Experimental Results
The authors validate the framework through simulation studies and an application to resting-state fMRI data:

• Simulations: Using VAR(1) and VMA(3) processes, the method demonstrates that the estimated confidence regions achieve coverage probabilities close to the nominal level (e.g., 95%) as sample size increases. The proposed method (deCGLASSO) exhibits significantly higher statistical power in detecting conditional edges compared to the benchmark nodewise regression with partial spectral coherence (NWR-PSC) method, particularly in smaller sample sizes. The FDR is controlled near the desired level (0.05), with minor exceedances that diminish as $n$ increases.
• fMRI Application: Applied to Human Connectome Project data (86 brain regions, 1200 time points), deCGLASSO identified a denser and more extensive set of significant frequency-specific functional connections compared to NWR-PSC. At higher frequencies (e.g., $\omega = \pi/2$), the benchmark method failed to detect significant edges, whereas deCGLASSO identified connections within and across major functional networks (visual, somatomotor, default mode). At lower frequencies, both methods detected more edges, but deCGLASSO maintained a denser pattern of discovery.

Significance
The paper claims that its primary significance lies in providing the first rigorous, likelihood-based inference framework for sparse spectral precision matrices in high dimensions. By addressing the unique challenges of complex-valued data and temporal dependence, the method enables valid statistical testing of frequency-specific conditional dependencies. This allows researchers to distinguish stable conditional relationships from spurious ones in dynamic systems, such as neural networks or financial markets, where association structures vary across temporal scales. The work bridges the gap between high-dimensional estimation and formal statistical inference in the spectral domain.