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Geometricity and Embedding

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Similarity-Based Pattern Analysis and Recognition

Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

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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.

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Authors and Affiliations

  1. Department of Computer Science, University of York, York, UK

    Peng Ren, Furqan Aziz, Lin Han, Eliza Xu, Richard C. Wilson & Edwin R. Hancock

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  1. Peng Ren
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  2. Furqan Aziz
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  3. Lin Han
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  4. Eliza Xu
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  6. Edwin R. Hancock
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Correspondence to Edwin R. Hancock .

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  1. DAIS, Ca' Foscari University of Venice, Venezia Mestre, Italy

    Marcello Pelillo

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Ren, P., Aziz, F., Han, L., Xu, E., Wilson, R.C., Hancock, E.R. (2013). Geometricity and Embedding. In: Pelillo, M. (eds) Similarity-Based Pattern Analysis and Recognition. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-5628-4_6

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  • DOI: https://doi.org/10.1007/978-1-4471-5628-4_6

  • Publisher Name: Springer, London

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