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.

翻译：多模态网络群体出现在许多场景中，包括大规模多模态神经影像研究，这些研究同时捕获了数千名受试者的功能性和结构性神经影像数据。此类研究的一个主要问题是功能性脑连接与结构性脑连接如何相互关联，以及它们如何在群体中变化。我们开发了一种新型的主成分分析（PCA）框架，用于整合在多个受试者上测量的多模态无向网络。具体而言，我们将这些网络排列为半对称张量，其中每个张量切片是一个对称矩阵，代表单个受试者的网络。随后，我们提出了一种新颖的联合、整合半对称张量主成分分析（JisstPCA）模型，并配以高效的迭代算法，用于联合发现同一受试者群体中两个或多个网络的低秩表示。我们建立了我们分离的低秩网络因子以及共享群体因子向真实因子的一步统计收敛性，并给出了有限样本统计误差界。通过模拟研究和整合多受试者功能性与结构性脑连接的真实数据实例，我们展示了我们的方法在多模态网络群体中发现联合低秩结构的优势。