  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.
