Multivariate Analysis for Multiple Network Data via Semi-Symmetric Tensor PCA

Network data are commonly collected in a variety of applications, representing either directly measured or statistically inferred connections between subjects or features of interest. In an increasing number of domains, these networks are collected over time, such as repeated interactions between users of a social media platform, or across multiple subjects, such as in multi-subject neuroimaging studies. When analyzing multiple large networks, dimensionality reduction techniques are often used to embed networks in a more tractable low-dimensional space. To this end, we develop a framework for principal components analysis (PCA) on collections of networks via a specialized tensor decomposition, termed Semi-Symmetric Tensor PCA or SST-PCA, and analyze it theoretically. Notably, we show that SST-PCA achieves the same accuracy as classical matrix PCA, with error proportional to the square root of the number of vertices and not the number of edges as might be expected. Our framework inherits many of the strengths of classical PCA and is suitable for a wide range of unsupervised learning tasks, including identifying principal networks, isolating changepoints and outliers, and for characterizing the “variability network” of the most varying edges. Finally, we demonstrate the effectiveness of SST-PCA in simulation and on an example from empirical social studies.