A Nonconvex Framework for Structured Dynamic Covariance Recovery
This work addresses the challenge of interpretable dynamic covariance estimation in neuroscience, offering a domain-specific solution with incremental improvements over prior methods.
The authors tackled the problem of estimating time-varying covariance matrices in high-dimensional data, specifically for functional neuroimaging, by proposing a nonconvex model that factors covariances into sparse spatial and smooth temporal components, and they demonstrated that their approach outperforms existing baselines in simulations and real data.
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.