A Nonconvex Framework for Structured Dynamic Covariance Recovery

Dynamic covariance models appear prominently in the analysis of time-series data and can provide fundamental insights into complex systems. They are important for applications ranging from computational finance and economics (Engle et al., 2019) to epidemiology (Fox and Dunson, 2015) and neuroscience (Foti and Fox, 2019). Our work is motivated by the study of dynamic functional brain network connectivity, defined as the time-indexed covariance of brain activity. Improving understanding of brain function is timely, as neurological and neuropsychiatric disorders ranging from Schizophrenia, Autism, and Alzheimer's to depression create burdens on affected individuals. To this end, understanding the variation of brain connectivity between individuals is believed to be a crucial step towards uncovering the mechanisms of neural information processing (Sakoğlu et al., 2010; Chang et al., 2016), with potentially transformative applications to understanding and treating neurological and neuropsychiatric disorders (Fox and Raichle, 2007; Calhoun et al., 2014). There are a variety of approaches for estimating dynamic covariances in the neuroscience literature. Discrete-state hidden Markov models construct simple explanations of brain connectivity in terms of recurring connectivity patterns (Vidaurre et al., 2017), and yet they fail to capture the smooth nature of brain dynamics (Shine et al., 2016a,b).