Hyperbolic Distance Matrices

Hyperbolic space is a natural setting for mining and visualizing data with hierarchical structure. In order to compute a hyperbolic embedding from comparison or similarity information, one has to solve a hyperbolic distance geometry problem. In this paper, we propose a unified framework to compute hyperbolic embeddings from an arbitrary mix of noisy metric and non-metric data. Our algorithms are based on semidefinite programming and the notion of a hyperbolic distance matrix, in many ways parallel to its famous Euclidean counterpart. A central ingredient we put forward is a semidefinite characterization of the hyperbolic Gramian -- a matrix of Lorentzian inner products. This characterization allows us to formulate a semidefinite relaxation to efficiently compute hyperbolic embeddings in two stages: first, we complete and denoise the observed hyperbolic distance matrix; second, we propose a spectral factorization method to estimate the embedded points from the hyperbolic distance matrix. We show through numerical experiments how the flexibility to mix metric and non-metric constraints allows us to efficiently compute embeddings from arbitrary data.