Dimensionality and Curvature Selection of Graph Embedding using Decomposed Normalized Maximum Likelihood Code-Length

Abstract

Graph embedding methods are effective techniques for representing nodes and their relations in a continuous space. Several studies try to embed graphs in constant curvature manifolds such as Euclidean, hyperbolic, and spherical space. It is critical how to select the best space for the graph embedding, as well as its dimensionality. In this study, we focus on the aforementioned constant curvature manifolds and aim at the dimensionality and curvature selection from the viewpoint of statistical model selection for latent variable models. Thereafter, we introduce universal latent variables models using wrapped normal distributions, which are the extension of Gaussian distribution for Riemannian manifolds. We then propose a novel methodology using decomposed normalized maximum likelihood code-length, which is based on the minimum description length principle. We empirically demonstrated the effectiveness of our method using both artificial and real-world datasets.