A Bayesian Approach to Hyperbolic Multi-Dimensional Scaling

Abstract

Recent studies have increasingly demonstrated that hyperbolic geometry confers many advantages for analyzing hierarchical structure in complex systems. However, available embedding methods do not give a precise metric for determining the dimensionality of the data, and do not vary curvature. These parameters are important for obtaining accurate, low dimensional, continuous descriptions of the data. To address this we develop a Bayesian formulation of Multi-Dimensional Scaling for embedding data in hyperbolic spaces that can fit for the optimal values of geometric parameters such as curvature and dimension. We propose a novel model of embedding uncertainty within this Bayesian framework which improves both performance and interpretability of the model. Because the method allows for variable curvature, it can also correctly embed Euclidean data using zero curvature, thus subsuming traditional Euclidean MDS models. We demonstrate that only a small amount of data is needed to constrain the geometry in our model and that the model is robust against false minima when scaling to large datasets. We apply our model to real world datasets and uncover new insights into their hierarchical structure derived from our geometric embeddings.