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Geometricity and Embedding

  • Peng Ren
  • Furqan Aziz
  • Lin Han
  • Eliza Xu
  • Richard C. Wilson
  • Edwin R. Hancock
Part of the Advances in Computer Vision and Pattern Recognition book series (ACVPR)

Abstract

In this chapter, we compare and contrast two approaches to the problem of embedding non-Euclidean data, namely geometric and structure preserving embedding. Under the first heading, we explore how spherical embedding can be used to embed data onto the surface of sphere of optimal radius. Here we explore both elliptic and hyperbolic geometries, i.e., positive and negative curvatures. Our results on synthetic and real data show that the elliptic embedding performs well under noisy conditions and can deliver low-distortion embeddings for a wide variety of datasets. Hyperbolic data seems to be much less common (at least in our datasets) and is more difficult to accurately embed. Under the second heading, we show how the Ihara zeta function can be used to embed hypergraphs in a manner which reflects their underlying relational structure. Specifically, we show how a polynomial characterization derived from the Ihara zeta function leads to an embedding which captures the prime cycle structure of the hypergraphs.

Keywords

Bipartite Graph Zeta Function Adjacency Matrix Geodesic Distance Laplacian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Peng Ren
    • 1
  • Furqan Aziz
    • 1
  • Lin Han
    • 1
  • Eliza Xu
    • 1
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

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