Abstract:

Flexible, yet interpretable, models for the second-order temporal structure are needed in scientific analyses of high-dimensional data. We develop a structured time-indexed covariance model for dynamic time-series data by factorizing covariances into sparse spatial and temporally smooth components. Traditionally, time-indexed covariance models without structure require a large sample size to be estimable. While the covariance factorization results in both domain interpretability and ease of estimation from the statistical perspective, the resulting optimization problem used to estimate the model components are nonconvex. 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 with a nontrivial statistical error for the proposed descent scheme and establish sample complexity guarantees for the estimator. As a motivating example, we consider the neuroscience application of estimation of dynamic brain connectivity. Empirical results using simulated and real brain imaging data illustrate that our approach outperforms existing baselines.