# [KDD 2020] Hyperbolic Distance Matrices

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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.

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