A Nonconvex Framework for Structured Dynamic Covariance Recovery

TL;DR

This paper introduces a nonconvex, interpretable model for estimating high-dimensional, time-varying covariance matrices in neuroimaging data, with proven convergence and superior empirical performance.

Contribution

It develops a novel two-stage nonconvex optimization framework with spectral initialization for structured covariance recovery, providing theoretical guarantees and improved results.

Findings

Outperforms existing methods on simulated data

Achieves linear convergence with statistical guarantees

Effective in real brain imaging applications

Abstract

We propose a flexible yet interpretable model for high-dimensional data with time-varying second order statistics, motivated and applied to functional neuroimaging data. Motivated by the neuroscience literature, we factorize the covariances into sparse spatial and smooth temporal components. While this factorization results in both parsimony and domain interpretability, the resulting estimation problem is nonconvex. To this end, we design a two-stage optimization scheme with a carefully tailored spectral initialization, combined with iteratively refined alternating projected gradient descent. We prove a linear convergence rate up to a nontrivial statistical error for the proposed descent scheme and establish sample complexity guarantees for the estimator. We further quantify the statistical error for the multivariate Gaussian case. Empirical results using simulated and real brain imaging data illustrate that our approach outperforms existing baselines.