Regularized Estimation of Sparse Spectral Precision Matrices

2024-01-20

AI Summary

To address key bottlenecks in high-dimensional frequency-domain analysis of time series—including poor scalability, difficulty handling component heterogeneity, and lack of non-asymptotic theoretical guarantees for estimating sparse spectral precision matrices (i.e., inverses of spectral density matrices)—this paper proposes the Complex Graphical Lasso (CGLASSO) and its adaptive extension (CAGLASSO). We introduce the first real-valued coordinate descent algorithm grounded in ring isomorphism, overcoming the complex-valued optimization challenges arising from the non-i.i.d. structure of the discrete Fourier transform (DFT). Moreover, we establish the first non-asymptotic error decomposition theory tailored to frequency-domain sparse estimation, rigorously characterizing both high-dimensional approximation and estimation errors. The proposed methods achieve superior statistical consistency, computational efficiency, and sparse structure recovery compared to state-of-the-art alternatives. Extensive simulations and applications to real neuroscience data empirically validate their advantages.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• Estimating sparse spectral precision matrices for high-dimensional time series
• Developing fast optimization algorithms for complex-valued graphical lasso
• Establishing non-asymptotic theory for approximation and estimation errors

Innovation

• Fast pathwise coordinate descent algorithms for optimization
• Adaptive penalization tailored to time series variability
• Non-asymptotic theory analyzing approximation and estimation errors