Asymmetric canonical correlation analysis of Riemannian and high-dimensional data

In this paper, we introduce a novel statistical model for the integrative analysis of Riemannian-valued functional data and high-dimensional data. We apply this model to explore the dependence structure between each subject's dynamic functional connectivity -- represented by a temporally indexed collection of positive definite covariance matrices -- and high-dimensional data representing lifestyle, demographic, and psychometric measures. Specifically, we employ a reformulation of canonical correlation analysis that enables efficient control of the complexity of the functional canonical directions using tangent space sieve approximations. Additionally, we enforce an interpretable group structure on the high-dimensional canonical directions via a sparsity-promoting penalty. The proposed method shows improved empirical performance over alternative approaches and comes with theoretical guarantees. Its application to data from the Human Connectome Project reveals a dominant mode of covariation between dynamic functional connectivity and lifestyle, demographic, and psychometric measures. This mode aligns with results from static connectivity studies but reveals a unique temporal non-stationary pattern that such studies fail to capture.