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Generalization Error Bound for Hyperbolic Ordinal Embedding
[article]

2021
*
arXiv
*
pre-print

*Hyperbolic*

*ordinal*

*embedding*(HOE) represents entities as points in

*hyperbolic*space so that they agree as well as possible with given constraints in the form of entity i is more similar to entity j than ... The difficulty is that existing generalization error bound derivations for

*ordinal*

*embedding*based on the Gramian matrix do not work in HOE, since

*hyperbolic*space is not inner-product space. ... ., 2019; Tabaghi & Dokmanic, 2020) have proposed

*ordinal*

*embedding*methods using

*hyperbolic*space for hierarchical data, which we call

*hyperbolic*

*ordinal*

*embedding*(HOE) in this paper. ...

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Hyperbolic Distance Matrices
[article]

2020
*
arXiv
*
pre-print

In order to compute a

arXiv:2005.08672v2
fatcat:vkm26ds4yvf3lku6qtl3ofnt7q
*hyperbolic**embedding*from comparison or similarity information, one has to solve a*hyperbolic*distance geometry problem. ... ; second, we propose a spectral factorization method to estimate the*embedded*points from the*hyperbolic*distance matrix. ... ACKNOWLEDGEMENT We thank Lav Varshney for bringing our attention to*hyperbolic*geometry and for the numerous discussions about the manuscript. ...##
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Neural Embeddings of Graphs in Hyperbolic Space
[article]

2017
*
arXiv
*
pre-print

We present a new concept that exploits these recent insights and propose learning neural

arXiv:1705.10359v1
fatcat:uzfzvsqaijcfxezywmyprjaz4q
*embeddings*of graphs in*hyperbolic*space. ... However, recent work has shown that the appropriate isometric space for*embedding*complex networks is not the flat Euclidean space, but negatively curved,*hyperbolic*space. ... To perform backpropagation it is easiest to work in natural*hyperbolic*co-*ordinates*on the disk and map back to Euclidean co-*ordinates*only at the end. ...##
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Geometry of Similarity Comparisons
[article]

2021
*
arXiv
*
pre-print

This leads to a lower bound on the Euclidean and spherical

arXiv:2006.09858v4
fatcat:ph7qemzarnagpi662sbawljxfq
*embedding*dimension of what we term similarity graphs. ... Many data analysis problems can be cast as distance geometry problems in space forms – Euclidean, spherical, or*hyperbolic*spaces. ... These results motivate the study of*ordinal*capacity.*Hyperbolicity*of Trees*Hyperbolic*spaces are space forms that offer small distortion when*embedding*trees [43, 44] . ...##
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Groups possessing extensive hierarchical decompositions

2010
*
Bulletin of the London Mathematical Society
*

We show that for each countable

doi:10.1112/blms/bdq045
fatcat:rbhxyhrlkzfxzk73zirfaqio6m
*ordinal*α, there is a countable group that is in Kropholler's class which does not appear until the α+1st stage of the hierarchy. ... ., a natural filtration indexed by the*ordinals*. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. ... SQ-universality and variations on the Rips' complex for*hyperbolic*groups We begin with an observation on the classical Higman, Neumann and Neumann*embedding*theorem. Lemma 3.1. ...##
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Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds
[article]

2020
*
arXiv
*
pre-print

In this paper we study existence and lack thereof of closed

arXiv:2005.11256v2
fatcat:bitf6w4wajfpnempsogwi4lbka
*embedded*orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable*hyperbolic*manifolds. ... Let N → X be an*embedded*orientable totally geodesic*hyperbolic*3-manifold. ... The four reflections in the co-*ordinate*planes of R 4 can be taken as generators of this (Z/2Z) 4 group of isometries. ...##
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Page 864 of American Journal of Mathematics Vol. 74, Issue 4
[page]

1952
*
American Journal of Mathematics
*

Wintner, “On the

*embedding*of*hyperbolic*line elements; a correc- tion,” ibid., vol. 74 (1952), p. 264. E. ... Cinquini-Cibrario, “Sopra la teoria delle caratteristiche per i sistemi di equazioni quasi-lineari alle derivate parziali del primo*ordine*,” Annali della Scuola Normale Superiore de Pisa, ser. 3, vol. ...##
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HEAT: Hyperbolic Embedding of Attributed Networks
[article]

2019
*
arXiv
*
pre-print

To fill this gap, we introduce HEAT (

arXiv:1903.03036v2
fatcat:76jvo6jgp5ejvhmakhap7v4mby
*Hyperbolic**Embedding*of ATributed networks), the first method for*embedding*attributed networks to a*hyperbolic*space. ... An emerging approach is*embedding*these networks into*hyperbolic*space because it can naturally represent a network's hierarchical structure. ... We say that point x ∈ R n:1 has time co-*ordinate*x 0 i and spacial coordinates x k i for k = 1, 2, ..., n. ...##
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Characteristics and existence of isometric embeddings

1983
*
Duke mathematical journal
*

The linearized isometric

doi:10.1215/s0012-7094-83-05040-8
fatcat:fed6qkm3prgxfk4bhygdnypfey
*embedding*system is accordingly strictly .*hyperbolic*or real principal type. ... In local co-*ordinates*, for Thus, to solve the linearized isometric*embedding*system, we must solve a system of first order partial differential equations. ... On the other hand if u 0 defines a strictly*hyperbolic*operator P0, then any u Dr(uo) will also define a strictly*hyperbolic*operator. ...##
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Almost continuous extension for taut foliations
[article]

2002
*
arXiv
*
pre-print

A taut foliation of a

arXiv:math/0008113v2
fatcat:okh4uwtqdbhx5c4dyr3gryybd4
*hyperbolic*3-manifold has the continuous extension property for leaves in almost every direction; that is, for each leaf of the universal cover of the foliation and almost every geodesic ... Thus λ\B dπ • i(x) α dvol λ < ∞ In particular, using spherical co-*ordinates*on λ, we can conclude that for almost every geodesic ray γ ⊂ λ emanating from p, ∞ c dπ(γ(t)) dt α e t dt < ∞ so in particular ... That is, there is an ǫ so that there is a quasi-isometric*embedding*I : λ × [−ǫ, ǫ] → H 3 such that I(p, 0) = i( * ). That the geometry of the*embedding*is bounded follows from the compactness of M . ...##
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Embeddings of Gromov Hyperbolic Spaces
[chapter]

2011
*
Selected Works of Oded Schramm
*

Another

doi:10.1007/978-1-4419-9675-6_10
fatcat:ixzar3aelvfaxixl6axe4x6meu
*embedding*theorem states that any 8-*hyperbolic*metric space embeds isometrically into a complete geodesic 8-*hyperbolic*space. ... Another*embedding*theorem presented here says that any 8-*hyperbolic*metric space X embeds isometrically in a complete geodesic 8-*hyperbolic*metric space. ... The present study of Gromov*hyperbolic*spaces was partly motivated by the work of Phil Bowers and Ken Stephenson on circle packings whose nerve is a planar*hyperbolic*graph. ...##
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Embeddings of Gromov hyperbolic spaces

2000
*
Geometric and Functional Analysis
*

Another

doi:10.1007/s000390050009
fatcat:uty6t5222jcj7kbidv2wyq2oai
*embedding*theorem states that any 8-*hyperbolic*metric space embeds isometrically into a complete geodesic 8-*hyperbolic*space. ... Another*embedding*theorem presented here says that any 8-*hyperbolic*metric space X embeds isometrically in a complete geodesic 8-*hyperbolic*metric space. ... The present study of Gromov*hyperbolic*spaces was partly motivated by the work of Phil Bowers and Ken Stephenson on circle packings whose nerve is a planar*hyperbolic*graph. ...##
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Regularising the Ricci Flow Embedding
[chapter]

2010
*
Lecture Notes in Computer Science
*

This paper concerns the analysis of patterns that are specified in terms of non-Euclidean dissimilarity or proximity rather than

doi:10.1007/978-3-642-14980-1_57
fatcat:tupnxbiyg5ewtdl6sgk7pkdsau
*ordinal*values. ... This is achieved by representing the data using a graph, and evolving the manifold*embedding*of the graph using Ricci flow. ... Let y u be the*embedded*co-*ordinates*of the node u ∈ V and Y = (y 1 |...|y |V | ) be the matrix with the*embedded*co-*ordinates*as columns. ...##
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Almost continuous extension for taut foliations

2001
*
Mathematical Research Letters
*

A taut foliation F of a

doi:10.4310/mrl.2001.v8.n5.a5
fatcat:4dv33m7etjezphvjvuwemqu7kq
*hyperbolic*3-manifold M has the continuous extension property for leaves in almost every direction; that is, for each leaf λ of F and almost every geodesic ray γ in λ the limit ... Thus λ\B dπ • i(x) α dvol λ < ∞ In particular, using spherical co-*ordinates*on λ, we can conclude that for almost every geodesic ray γ ⊂ λ emanating from p, ∞ c dπ(γ(t)) dt α e t dt < ∞ so in particular ... That is, there is an so that there is a quasi-isometric*embedding*I : λ × [− , ] → H 3 such that I(p, 0) = i( * ). That the geometry of the*embedding*is bounded follows from the compactness of M . ...##
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Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in S^4
[article]

2021
*
arXiv
*
pre-print

In this paper we prove that certain

arXiv:2109.01687v1
fatcat:m7ygefpmsnavzi77pleiwpmjpi
*hyperbolic*link complements of 2-tori in S^4 do not contain closed*embedded*totally geodesic*hyperbolic*3-manifolds. ... The focus of this paper is obstructing the*embedding*of closed*hyperbolic*3-manifolds in S 4 via*embeddings*in*hyperbolic*link complements of 2-tori in S 4 . ... The four reflections in the co-*ordinate*planes of R 4 can be taken as generators of this (Z/2Z) 4 group of isometries. ...
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