A Bayesian Approach to Hyperbolic Embeddings

Recent studies have increasingly demonstrated that hyperbolic geometry confers many advantages for analyzing hierarchical structure in complex systems. However, available embedding methods for hyperbolic spaces typically operate at fixed dimension (usually 2 or 3), do not vary curvature, and require knowledge of network connections between data points. To address these problems, 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, physics based 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 show how the estimated geometry can be used to derive a new hierarchical clustering algorithm, and demonstrate its effectiveness for inferring latent hierarchical structure in the data. We demonstrate the capabilities of the model by applying it to a variety of biological datasets, uncovering hidden hierarchical relationships in datasets relating to aging and the COVID genome