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Theoretical and Applied Aspects of the Self-Organizing Maps

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Advances in Self-Organizing Maps and Learning Vector Quantization

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 428))

Abstract

The Self-Organizing Map (SOM) is widely used, easy to implement, has nice properties for data mining by providing both clustering and visual representation. It acts as an extension of the k-means algorithm that preserves as much as possible the topological structure of the data. However, since its conception, the mathematical study of the SOM remains difficult and has be done only in very special cases. In WSOM 2005, Jean-Claude Fort presented the state of the art, the main remaining difficulties and the mathematical tools that can be used to obtain theoretical results on the SOM outcomes. These tools are mainly Markov chains, the theory of Ordinary Differential Equations, the theory of stability, etc. This article presents theoretical advances made since then. In addition, it reviews some of the many SOM algorithm variants which were defined to overcome the theoretical difficulties and/or adapt the algorithm to the processing of complex data such as time series, missing values in the data, nominal data, textual data, etc.

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Notes

  1. 1.

    EM is the standard algorithm for mixture models.

  2. 2.

    A kernel is a particular case of symmetric similarity such that K is a symmetric matrix, semi-definite positive with \(K(x_i,x_i)=0\) and satisfies the following positive constraint

    $$\forall \,M>0, \ \forall \,(x_i)_{i=1,\ldots ,M}\in \mathcal {X},\ \forall \,(\alpha _i)_{i=1,\ldots ,M},\qquad \sum _{i,j} \alpha _i\alpha _j K(x_i,x_j) \ge 0.$$

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Authors and Affiliations

  1. SAMM - Université Paris 1 Panthéon-Sorbonne, 90, rue de Tolbiac, 75013, Paris, France

    Marie Cottrell, Madalina Olteanu & Fabrice Rossi

  2. INRA, UR 0875 MIAT, BP 52627, 31326, Castanet Tolosan, France

    Nathalie Villa-Vialaneix

Authors
  1. Marie Cottrell
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  2. Madalina Olteanu
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  3. Fabrice Rossi
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  4. Nathalie Villa-Vialaneix
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Correspondence to Marie Cottrell or Nathalie Villa-Vialaneix .

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Editors and Affiliations

  1. Department of Statistics, Rice University, Houston, Texas, USA

    Erzsébet Merényi

  2. Department of Electrical and Computer En, Air Force Institute of Technology, Wright-Patterson AFB, Ohio, USA

    Michael J. Mendenhall

  3. Applied Physics, Rice University, Houston, Texas, USA

    Patrick O'Driscoll

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Cottrell, M., Olteanu, M., Rossi, F., Villa-Vialaneix, N. (2016). Theoretical and Applied Aspects of the Self-Organizing Maps. In: Merényi, E., Mendenhall, M., O'Driscoll, P. (eds) Advances in Self-Organizing Maps and Learning Vector Quantization. Advances in Intelligent Systems and Computing, vol 428. Springer, Cham. https://doi.org/10.1007/978-3-319-28518-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-28518-4_1

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