Dimensionality Selection of Hyperbolic Graph Embeddings using Decomposed Normalized Maximum Likelihood Code-Length

Ryo Yuki, Yuichi Ike, Kenji Yamanishi

Abstract: Graph embedding methods are effective techniques for representing nodes and their relations in a continuous space. Specifically, hyperbolic space is more effective for embedding graphs with tree-like structures than Euclidean one. Then it is critical how to select the best dimensionality of the hyperbolic space where a graph is embedded. This is because we cannot distinguish nodes well with too low a dimensionality, whereas the embedded relations are affected by irregularities of data with too high a dimensionality. We consider this problem from the view of statistical model selection for latent variable models. We then propose a novel methodology for dimensionality selection on the basis of the minimum description length principle. The key idea is to make the latent variable model of hyperbolic embeddings and to employ the decomposed normalized maximum likelihood code-length as an evaluation criterion. We empirically demonstrate the effectiveness of our method through synthetic and real datasets.