Factor Analysis on a Graph

Graph is a common way to represent relationships among a set of objects in a variety of application areas of machine learning. We consider the case that the input data is not only a graph but also numerical features in which one of the given features corresponds to a node in the graph. Then, the primary importance is often in understanding interactions on the graph nodes which effect on covariance structure of the numerical features. We propose a Gaussian based analysis which is a combination of graph constrained covariance matrix estimation and factor analysis (FA). We show that this approach, called graph FA, has desirable interpretability. In particular, we prove the connection between graph FA and a graph node clustering based on a perspective of kernel method. This connection indicates that graph FA is effective not only on the conventional noise-reduction explanation of the observation by FA but also on identifying important subgraphs. The experiments on synthetic and real-world datasets demonstrate the effectiveness of the approach.