Regularized Estimation of Sparse Spectral Precision Matrices

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.