Low Distortion Delaunay Embedding of Trees in Hyperbolic Plane
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Any tree can be embedded with arbitrary low distortion into the Poincaré disk (with only 2 dimensions), whereas this is not true for Euclidean spaces even when an unbounded number of dimensions is allowed.
“Conversely, any tree can be embedded with arbitrary low distortion into the Poincar´ e disk (with only 2 dimensions), whereas this is not true for Euclidean spaces even when an unbounded number of dimensions is allowed (Sarkar, 2011; De Sa et al., 2018).”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Sarkar, 2011
The indexed results for distortion, embedding, and hyperbolic space returned only abstract and bibliographic text, not a passage from the paper’s full text that can serve as evidence under this rubric. No eligible passage establishes the claim about arbitrarily low distortion in the Poincaré disk or the comparison with Euclidean spaces. (LLM, conf=0.62)
Experimental — accuracy not yet measuredHyperbolic spaces provably model tree-like structures better than Euclidean geometry.
“Following prior work, we first advocate for using hyperbolic spaces which provably model tree-like structures better than Euclidean geometry.”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Sarkar, 2011
Searches for Euclidean geometry, trees, and distortion found no substantive passage from the paper’s body. The indexed result contains an abstract describing low-distortion embeddings in the hyperbolic plane, but no body-text passage establishing the claimed comparison with Euclidean geometry; the abstract cannot decide the verdict under the evidence rule. (LLM, conf=0.70)
Experimental — accuracy not yet measuredThe exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric.
“This exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric 3 (Gromov, 1987; Bowditch, 2006; Sarkar, 2011).”
From Hyperbolic Entailment Cones for Learning Hierarchical Embeddings (2018) · cites this paper as Sarkar, 2011
The indexed matches contain only an Abstract and bibliographic metadata, not a passage from the paper’s full text that can serve as evidence under the stated rule. Searches for “exponential,” “growth,” and weighted-tree embedding found no qualifying full-text passage establishing the attributed explanation. (LLM, conf=0.72)
Experimental — accuracy not yet measured
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Cites
Gromov, M. (1987) · Mathematical Sciences Research Institute Publications · doi:10.1007/978-1-4613-9586-7_3 · Cited by 1,817 · see citing works
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Hyperbolic Entailment Cones for Learning Hierarchical Embeddings
Ganea, O.-E., Bécigneul, G., & Hofmann, T. (2018) · arXiv · doi:10.48550/arxiv.1804.01882 · Cited by 115 · see citing works
“This exponential growth property enables hyperbolic spaces to embed any weighted tree while almost preserving their metric 3 (Gromov, 1987; Bowditch, 2006; Sarkar, 2011).”
Background
- Authors
- Sarkar, R.
- Affiliations
- Freie Universität Berlin (DE)
- Corresponding author
- Sarkar, Rik
- ORCID
- Sarkar, Rik
- Venue
- Lecture Notes in Computer Science
- Publisher
- Springer Berlin Heidelberg
- Published
- 2012
- Pages
- 355-366
- ISSN
- 0302-9743 (ISSN-L)
- 1611-3349
- OpenAlex
- W2199357291
- Cited by
- 168
- References
- 1
- Citations by year
- 2026: 11 · 2025: 20 · 2024: 34 · 2023: 19 · 2022: 16 · 2021: 25 · 2020: 23 · 2019: 14 · 2018: 3 · 2016: 1 · 2014: 2
- FWCI
- 0.73
- Type
- conference paper
- Language
- en
- Open access
- Licence
- Locations
- Lecture notes in computer science (publishedVersion)
- gdea - Graph Drawing E-print Archive (GDEA) (submittedVersion)
- Edinburgh Research Explorer (University of Edinburgh) (submittedVersion, other-oa)
- Funders
- Deutsche Forschungsgemeinschaft (GRK1408)
- Topics
- Physical Sciences › Computer Science › Computer Graphics and Computer-Aided Design › Computational Geometry and Mesh Generation
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Complexity and Algorithms in Graphs
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Keywords
- tree embedding, hyperbolic plane, Delaunay graph, distance distortion, metric embeddings, weighted trees, hyperbolic distance, greedy routing, minimum spanning tree, discrete metric
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