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Stable Geodesic Update on Hyperbolic Space and its Application to Poincare Embeddings [article]

Yosuke Enokida, Atsushi Suzuki, Kenji Yamanishi
2018 arXiv   pre-print
A hyperbolic space has been shown to be more capable of modeling complex networks than a Euclidean space. This paper proposes an explicit update rule along geodesics in a hyperbolic space.  ...  Experimental results demonstrate the good performance of our algorithm in the \Poincare embeddings of knowledge base data.  ...  Hyperbolic Space and its Geodesics In this section, we introduce a hyperbolic space and its geometry.  ... 
arXiv:1805.10487v1 fatcat:2khszzebkjcwfhcm44jlerne5m

Computing shortest homotopic cycles on polyhedral surfaces with hyperbolic uniformization metric

Miao Jin, Ning Ding, Yang Yang
2013 Computer-Aided Design  
The problem of computing shortest homotopic cycles on a surface has various applications in computational geometry and graphics.  ...  or cycles on a surface.  ...  Acknowledgments The authors would like to thank Dr. Xianfeng Gu for the discussion and the anonymous reviewers for their valuable suggestions.  ... 
doi:10.1016/j.cad.2012.07.015 fatcat:trqhzpfcqbg5peqjpyl3vf5hkq

From Node Embedding To Community Embedding : A Hyperbolic Approach [article]

Thomas Gerald, Hadi Zaatiti, Hatem Hajri, Nicolas Baskiotis, Olivier Schwander
2020 arXiv   pre-print
We illustrate the usefulness of this framework through several experiments on real-world social networks and comparisons with ComE and recent hyperbolic-based classification approaches.  ...  Our proposed approach combines hyperbolic embeddings with Riemannian K-means or Riemannian mixture models to perform community detection.  ...  ., 2014] and then update embeddings for selected nodes based on O 2 . 3. Update embeddings to fit the GMM based on O 3 .  ... 
arXiv:1907.01662v2 fatcat:6ugigvqaovhm5bvhgkplbgt5pe

Hyperbolic Deep Neural Networks: A Survey [article]

Wei Peng, Tuomas Varanka, Abdelrahman Mostafa, Henglin Shi, Guoying Zhao
2021 arXiv   pre-print
Such a hyperbolic neural architecture potentially leads to drastically compact model withmuch more physical interpretability than its counterpart in Euclidean space.  ...  generalization of the leading deep approaches to the Hyperbolic space.  ...  We also want to thank Emile Mathieu, from University of Oxford, for the explanation regarding the gyroplane layer in their Poincaré Variational Auto-Encoder.  ... 
arXiv:2101.04562v3 fatcat:yqj4zohrqjbplpsdy5f5uglnbu

Highly Scalable and Provably Accurate Classification in Poincare Balls [article]

Eli Chien, Chao Pan, Puoya Tabaghi, Olgica Milenkovic
2021 arXiv   pre-print
Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform required learning tasks.  ...  For hierarchical data, the space of choice is a hyperbolic space since it guarantees low-distortion embeddings for tree-like structures.  ...  Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform required learning tasks.  ... 
arXiv:2109.03781v3 fatcat:hpicmn6zyjfxblf5syfpofm5em

The hyperbolic geometry of financial networks [article]

Martin Keller-Ressel, Stephanie Nargang
2020 arXiv   pre-print
This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincar\'e disc model of hyperbolic geometry.  ...  We show that the latent dimensions of 'popularity' and 'similarity' in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system.  ...  Hyperbolic Embedding and Centering Embedding into Hyperbolic Space Network embedding methods aim to find -for each network node b i -latent coordinates x i in a geometric model space G, such that the  ... 
arXiv:2005.00399v2 fatcat:du7sz6nsfrgvtadihcyugab2ry

Computing Teichmuller Shape Space

Miao Jin, Wei Zeng, Feng Luo, Xianfeng Gu
2009 IEEE Transactions on Visualization and Computer Graphics  
A curve in the Teichmüller space represents a deformation process from one class to the other.  ...  In this work, we apply Teichmüller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic, invariant under the rigid motions and scalings, insensitive to resolutions.  ...  Figure 6 6 Figure 6(b) shows the embedding of fundamental domain of vase model onto the Poincaré disk with its Hyperbolic Uniformization metric.  ... 
doi:10.1109/tvcg.2008.103 pmid:19282555 fatcat:52q7xg2utvbknnklvh3ecixuli

Poincare Maps for Analyzing Complex Hierarchies in Single-Cell Data [article]

Anna Klimovskaia, David Lopez-Paz, Léon Bottou, Maximilian Nickel
2019 bioRxiv   pre-print
To overcome this fundamental representation issue we propose Poincaré maps, a method harnessing the power of hyperbolic geometry into the realm of single-cell data analysis.  ...  Often understood as a continuous extension of trees, hyperbolic geometry enables the embedding of complex hierarchical data in as few as two dimensions and well-preserves distances between points in the  ...  Acknowledgments We would like to thank Ioana Sandu and Will Macnair for valuable discussions.  ... 
doi:10.1101/689547 fatcat:rgr6kqzaljg33ocqepqfguxdvu

Representation Tradeoffs for Hyperbolic Embeddings [article]

Christopher De Sa, Albert Gu, Christopher Ré, Frederic Sala
2018 arXiv   pre-print
We provide upper and lower bounds that allow us to characterize the precision-dimensionality tradeoff inherent in any hyperbolic embedding.  ...  To embed general metric spaces, we propose a hyperbolic generalization of multidimensional scaling (h-MDS).  ...  Lines and geodesics There are two types of geodesics (shortest paths) in the Poincaré disk model of hyperbolic space: segments of circles that are orthogonal to the disk surface, and disk diameters [3  ... 
arXiv:1804.03329v2 fatcat:6cqhbstyffh7lokisclwnjof4m

Studying ventricular abnormalities in mild cognitive impairment with hyperbolic Ricci flow and tensor-based morphometry

Jie Shi, Cynthia M. Stonnington, Paul M. Thompson, Kewei Chen, Boris Gutman, Cole Reschke, Leslie C. Baxter, Eric M. Reiman, Richard J. Caselli, Yalin Wang
2015 NeuroImage  
Here we describe a novel ventricular morphometry system based on the hyperbolic Ricci flow method and tensor-based morphometry (TBM) statistics.  ...  Our system generates a one-to-one diffeomorphic mapping between ventricular surfaces with consistent boundary matching conditions.  ...  Poincaré disk model As the hyperbolic space cannot be realized in ℝ 3 , we use the Poincaré disk model to visualize it.  ... 
doi:10.1016/j.neuroimage.2014.09.062 pmid:25285374 pmcid:PMC4252650 fatcat:35wpd2c4dnfsxjdpp3ogf2f3pa

Conformal invariants for multiply connected surfaces: Application to landmark curve-based brain morphometry analysis

Jie Shi, Wen Zhang, Miao Tang, Richard J. Caselli, Yalin Wang
2017 Medical Image Analysis  
Our algorithm provides a stable method to compute the shape index values in the 2D (Poincaré Disk) parameter domain. The proposed shape indices are succinct, intrinsic and informative.  ...  With the surface Ricci flow method, we can conformally map a multiply connected surface to the Poincaré disk.  ...  Hoffmann-La Roche Ltd and its affiliated  ... 
doi:10.1016/j.media.2016.09.001 pmid:27639215 pmcid:PMC5099092 fatcat:qczplwszpzdppanhoom5wiwvaa

The hyperbolic geometry of financial networks

Martin Keller-Ressel, Stephanie Nargang
2021 Scientific Reports  
Using two different hyperbolic embedding methods, hydra+ and Mercator, this allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the  ...  Based on our analysis we argue that embeddings into hyperbolic geometry can be used to monitor structural change in financial networks and are able to distinguish between changes in systemic relevance  ...  However, due to the asymmetric distribution of banks within the Poincaré disc ( Fig. 4A ) for the hydra+ embedding, we calculate its geodesic polar coordinates (r i , θ i ) with respect to the network  ... 
doi:10.1038/s41598-021-83328-4 pmid:33637827 fatcat:ahp66g3vfzfkfir5oh2abgrfgq

Discrete Surface Ricci Flow

M. Jin, J. Kim, F. Luo, X. Gu
2008 IEEE Transactions on Visualization and Computer Graphics  
Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process.  ...  The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy.  ...  The embedding of its canonical fundamental domain in hyperbolic space has 4g sides, Fig. 5 (b) in Poincaré disk).  ... 
doi:10.1109/tvcg.2008.57 pmid:18599915 fatcat:z5zka2er35corovmdzazoxdpyy

Discrete Surface Ricci Flow [chapter]

Wei Zeng, Xianfeng David Gu
2013 SpringerBriefs in Mathematics  
Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process.  ...  The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy.  ...  The embedding of its canonical fundamental domain in hyperbolic space has 4g sides, Fig. 5 (b) in Poincaré disk).  ... 
doi:10.1007/978-1-4614-8781-4_4 fatcat:begfg75rhzdw7lsz6mgneufm34

Hyperbolic Harmonic Mapping for Constrained Brain Surface Registration

Rui Shi, Wei Zeng, Zhengyu Su, Hanna Damasio, Zhonglin Lu, Yalin Wang, Shing-Tung Yau, Xianfeng Gu
2013 2013 IEEE Conference on Computer Vision and Pattern Recognition  
It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture.  ...  This work conquer this problem by changing the Riemannian metric on the target surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints  ...  Hyperbolic embedding of M and N on Poincaré disk. (c). Decompose M and N into multiple pants, and each pant further decomposed to 2 hyperbolic hexagons. (d).  ... 
doi:10.1109/cvpr.2013.327 dblp:conf/cvpr/ShiZSDLWYG13 fatcat:56nay4zewbakhpiiyxv5cxp65y
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