Magnification factors for the SOM and GTM algorithms

Christopher M. Bishop, Markus Svens'en, Christopher K. I. Williams

    Research output: Contribution to conferencePaper

    Abstract

    Magnification factors specify the extent to which the area of a small patch of the latent (or `feature') space of a topographic mapping is magnified on projection to the data space, and are of considerable interest in both neuro-biological and data analysis contexts. Previous attempts to consider magnification factors for the self-organizing map (SOM) algorithm have been hindered because the mapping is only defined at discrete points (given by the reference vectors). In this paper we consider the batch version of SOM, for which a continuous mapping can be defined, as well as the Generative Topographic Mapping (GTM) algorithm of Bishop et al. (1997) which has been introduced as a probabilistic formulation of the SOM. We show how the techniques of differential geometry can be used to determine magnification factors as continuous functions of the latent space coordinates. The results are illustrated here using a problem involving the identification of crab species from morphological data.
    Original languageEnglish
    Publication statusPublished - 1997
    EventProceedings 1997 Workshop on Self-Organizing Maps - Helsinki University of Technology, Finland
    Duration: 4 Jun 19976 Jun 1997

    Workshop

    WorkshopProceedings 1997 Workshop on Self-Organizing Maps
    CountryFinland
    CityHelsinki University of Technology
    Period4/06/976/06/97

    Fingerprint

    topographic mapping
    biological analysis
    crab
    geometry

    Keywords

    • Magnification factors
    • topographic mapping
    • discrete
    • neuro-biological
    • Generative Topographic Mapping
    • differential geometry
    • morphological

    Cite this

    Bishop, C. M., Svens'en, M., & Williams, C. K. I. (1997). Magnification factors for the SOM and GTM algorithms. Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland.
    Bishop, Christopher M. ; Svens'en, Markus ; Williams, Christopher K. I. / Magnification factors for the SOM and GTM algorithms. Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland.
    @conference{3132a66861104472a01ec75427fd4713,
    title = "Magnification factors for the SOM and GTM algorithms",
    abstract = "Magnification factors specify the extent to which the area of a small patch of the latent (or `feature') space of a topographic mapping is magnified on projection to the data space, and are of considerable interest in both neuro-biological and data analysis contexts. Previous attempts to consider magnification factors for the self-organizing map (SOM) algorithm have been hindered because the mapping is only defined at discrete points (given by the reference vectors). In this paper we consider the batch version of SOM, for which a continuous mapping can be defined, as well as the Generative Topographic Mapping (GTM) algorithm of Bishop et al. (1997) which has been introduced as a probabilistic formulation of the SOM. We show how the techniques of differential geometry can be used to determine magnification factors as continuous functions of the latent space coordinates. The results are illustrated here using a problem involving the identification of crab species from morphological data.",
    keywords = "Magnification factors, topographic mapping, discrete, neuro-biological, Generative Topographic Mapping, differential geometry, morphological",
    author = "Bishop, {Christopher M.} and Markus Svens'en and Williams, {Christopher K. I.}",
    year = "1997",
    language = "English",
    note = "Proceedings 1997 Workshop on Self-Organizing Maps ; Conference date: 04-06-1997 Through 06-06-1997",

    }

    Bishop, CM, Svens'en, M & Williams, CKI 1997, 'Magnification factors for the SOM and GTM algorithms' Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland, 4/06/97 - 6/06/97, .

    Magnification factors for the SOM and GTM algorithms. / Bishop, Christopher M.; Svens'en, Markus; Williams, Christopher K. I.

    1997. Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland.

    Research output: Contribution to conferencePaper

    TY - CONF

    T1 - Magnification factors for the SOM and GTM algorithms

    AU - Bishop, Christopher M.

    AU - Svens'en, Markus

    AU - Williams, Christopher K. I.

    PY - 1997

    Y1 - 1997

    N2 - Magnification factors specify the extent to which the area of a small patch of the latent (or `feature') space of a topographic mapping is magnified on projection to the data space, and are of considerable interest in both neuro-biological and data analysis contexts. Previous attempts to consider magnification factors for the self-organizing map (SOM) algorithm have been hindered because the mapping is only defined at discrete points (given by the reference vectors). In this paper we consider the batch version of SOM, for which a continuous mapping can be defined, as well as the Generative Topographic Mapping (GTM) algorithm of Bishop et al. (1997) which has been introduced as a probabilistic formulation of the SOM. We show how the techniques of differential geometry can be used to determine magnification factors as continuous functions of the latent space coordinates. The results are illustrated here using a problem involving the identification of crab species from morphological data.

    AB - Magnification factors specify the extent to which the area of a small patch of the latent (or `feature') space of a topographic mapping is magnified on projection to the data space, and are of considerable interest in both neuro-biological and data analysis contexts. Previous attempts to consider magnification factors for the self-organizing map (SOM) algorithm have been hindered because the mapping is only defined at discrete points (given by the reference vectors). In this paper we consider the batch version of SOM, for which a continuous mapping can be defined, as well as the Generative Topographic Mapping (GTM) algorithm of Bishop et al. (1997) which has been introduced as a probabilistic formulation of the SOM. We show how the techniques of differential geometry can be used to determine magnification factors as continuous functions of the latent space coordinates. The results are illustrated here using a problem involving the identification of crab species from morphological data.

    KW - Magnification factors

    KW - topographic mapping

    KW - discrete

    KW - neuro-biological

    KW - Generative Topographic Mapping

    KW - differential geometry

    KW - morphological

    UR - http://www.cis.hut.fi/wsom97/progabstracts/19.html

    M3 - Paper

    ER -

    Bishop CM, Svens'en M, Williams CKI. Magnification factors for the SOM and GTM algorithms. 1997. Paper presented at Proceedings 1997 Workshop on Self-Organizing Maps, Helsinki University of Technology, Finland.