  Multi-modal populations of networks arise in many scenarios including in
large-scale multi-modal neuroimaging studies that capture both functional and
structural neuroimaging data for thousands of subjects. A major research
question in such studies is how functional and structural brain connectivity
are related and how they vary across the population. we develop a novel
PCA-type framework for integrating multi-modal undirected networks measured on
many subjects. Specifically, we arrange these networks as semi-symmetric
tensors, where each tensor slice is a symmetric matrix representing a network
from an individual subject. We then propose a novel Joint, Integrative
Semi-Symmetric Tensor PCA (JisstPCA) model, associated with an efficient
iterative algorithm, for jointly finding low-rank representations of two or
more networks across the same population of subjects. We establish one-step
statistical convergence of our separate low-rank network factors as well as the
shared population factors to the true factors, with finite sample statistical
error bounds. Through simulation studies and a real data example for
integrating multi-subject functional and structural brain connectivity, we
illustrate the advantages of our method for finding joint low-rank structures
in multi-modal populations of networks.
