A Coupled CP Decomposition for Principal Components Analysis of Symmetric Networks

In a number of application domains, one observes a sequence of network data; for example, repeated measurements between users interactions in social media platforms, financial correlation networks over time, or across subjects, as in multi-subject studies of brain connectivity. One way to analyze such data is by stacking networks into a third-order array or tensor. We propose a principal components analysis (PCA) framework for sequence network data, based on a novel decomposition for semi-symmetric tensors. We derive efficient algorithms for computing our proposed "Coupled CP" decomposition and establish estimation consistency of our approach under an analogue of the spiked covariance model with rates the same as the matrix case up to a logarithmic term. 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 meaningful changepoints or outliers across observations, and for characterizing the "variability network" of the most varying edges. Finally, we demonstrate the effectiveness of our proposal on simulated data and on examples from political science and financial economics. The proof techniques used to establish our main consistency results are surprisingly straight-forward and may find use in a variety of other matrix and tensor decomposition problems.