Hyperbolic distance matrices

Tabaghi, Puoya and Dokmanić, Ivan. (2020) Hyperbolic distance matrices. In: KDD '20: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. New York, pp. 1728-1738.

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Official URL: https://edoc.unibas.ch/81802/

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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.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Informatik > Data Analytics (Dokmanic)
UniBasel Contributors:Dokmanić, Ivan
Item Type:Conference or Workshop Item, refereed
Conference or workshop item Subtype:Conference Paper
Publisher:Association for Computing Machinery
Note:Publication type according to Uni Basel Research Database: Conference paper
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Last Modified:16 Feb 2021 11:35
Deposited On:16 Feb 2021 11:35

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