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Generalization Error Bound for Hyperbolic Ordinal Embedding [article]

Atsushi Suzuki, Atsushi Nitanda, Jing Wang, Linchuan Xu, Marc Cavazza, Kenji Yamanishi
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.  ... 
arXiv:2105.10475v1 fatcat:drx2h4ckuna45kqrbz72w6hppi

Hyperbolic Distance Matrices [article]

Puoya Tabaghi, Ivan Dokmanić
2020 arXiv   pre-print
In order to compute a 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.  ... 
arXiv:2005.08672v2 fatcat:vkm26ds4yvf3lku6qtl3ofnt7q

Neural Embeddings of Graphs in Hyperbolic Space [article]

Benjamin Paul Chamberlain, James Clough, Marc Peter Deisenroth
2017 arXiv   pre-print
We present a new concept that exploits these recent insights and propose learning neural 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.  ... 
arXiv:1705.10359v1 fatcat:uzfzvsqaijcfxezywmyprjaz4q

Geometry of Similarity Comparisons [article]

Puoya Tabaghi, Jianhao Peng, Olgica Milenkovic, Ivan Dokmanić
2021 arXiv   pre-print
This leads to a lower bound on the Euclidean and spherical 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] .  ... 
arXiv:2006.09858v4 fatcat:ph7qemzarnagpi662sbawljxfq

Groups possessing extensive hierarchical decompositions

T. Januszkiewicz, P. H. Kropholler, I. J. Leary
2010 Bulletin of the London Mathematical Society  
We show that for each countable 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.  ... 
doi:10.1112/blms/bdq045 fatcat:rbhxyhrlkzfxzk73zirfaqio6m

Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds [article]

Michelle Chu, Alan W. Reid
2020 arXiv   pre-print
In this paper we study existence and lack thereof of closed 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.  ... 
arXiv:2005.11256v2 fatcat:bitf6w4wajfpnempsogwi4lbka

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.  ... 

HEAT: Hyperbolic Embedding of Attributed Networks [article]

David McDonald, Shan He
2019 arXiv   pre-print
To fill this gap, we introduce HEAT (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.  ... 
arXiv:1903.03036v2 fatcat:76jvo6jgp5ejvhmakhap7v4mby

Characteristics and existence of isometric embeddings

Robert L. Bryant, Phillip A. Griffiths, Deane Yang
1983 Duke mathematical journal  
The linearized isometric 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.  ... 
doi:10.1215/s0012-7094-83-05040-8 fatcat:fed6qkm3prgxfk4bhygdnypfey

Almost continuous extension for taut foliations [article]

Danny Calegari
2002 arXiv   pre-print
A taut foliation of a 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 .  ... 
arXiv:math/0008113v2 fatcat:okh4uwtqdbhx5c4dyr3gryybd4

Embeddings of Gromov Hyperbolic Spaces [chapter]

M. Bonk, O. Schramm
2011 Selected Works of Oded Schramm  
Another 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.  ... 
doi:10.1007/978-1-4419-9675-6_10 fatcat:ixzar3aelvfaxixl6axe4x6meu

Embeddings of Gromov hyperbolic spaces

M. Bonk, O. Schramm
2000 Geometric and Functional Analysis  
Another 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.  ... 
doi:10.1007/s000390050009 fatcat:uty6t5222jcj7kbidv2wyq2oai

Regularising the Ricci Flow Embedding [chapter]

Weiping Xu, Edwin R. Hancock, Richard C. Wilson
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 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.  ... 
doi:10.1007/978-3-642-14980-1_57 fatcat:tupnxbiyg5ewtdl6sgk7pkdsau

Almost continuous extension for taut foliations

Danny Calegari
2001 Mathematical Research Letters  
A taut foliation F of a 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 .  ... 
doi:10.4310/mrl.2001.v8.n5.a5 fatcat:4dv33m7etjezphvjvuwemqu7kq

Totally geodesic hyperbolic 3-manifolds in hyperbolic link complements of tori in S^4 [article]

Michelle Chu, Alan W. Reid
2021 arXiv   pre-print
In this paper we prove that certain 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.  ... 
arXiv:2109.01687v1 fatcat:m7ygefpmsnavzi77pleiwpmjpi
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