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On the Quantization Error in SOM vs. VQ: A Critical and Systematic Study

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Advances in Self-Organizing Maps (WSOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5629))

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Abstract

The self-organizing map (SOM) is related to the classical vector quantization (VQ). Like in the VQ, the SOM represents a distribution of input data vectors using a finite set of models. In both methods, the quantization error (QE) of an input vector can be expressed, e.g., as the Euclidean norm of the difference of the input vector and the best-matching model. Since the models are usually optimized in the VQ so that the sum of the squared QEs is minimized for the given set of training vectors, a common notion is that it will be impossible to find models that produce a smaller rms QE. Therefore it has come as a surprise that in some cases the rms QE of a SOM can be smaller than that of a VQ with the same number of models and the same input data. This effect may manifest itself if the number of training vectors per model is on the order of small integers and the testing is made with an independent set of test vectors. An explanation seems to ensue from statistics. Each model vector in the VQ is determined as the average of those training vectors that are mapped into the same Voronoi domain as the model vector. On the contrary, each model vector of the SOM is determined as a weighted average of all of those training vectors that are mapped into the “topological” neighborhood around the corresponding model. The number of training vectors mapped into the neighborhood of a SOM model is generally much larger than that mapped into a Voronoi domain around a model in the VQ. Since the SOM model vectors are then determined with a significantly higher statistical accuracy, the Voronoi domains of the SOM are significantly more regular, and the resulting rms QE may then be smaller than in the VQ. However, the effective dimensionality of the vectors must also be sufficiently high.

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References

  1. Bação, O., Lobo, V., Painho, M.: Self-organizing maps as substitutes for k-means clustering. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3516, pp. 476–483. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  2. Cole, R.A., Muthusamy, Y., Fanty, M.A.: The ISOLET Spoken Letter Database, Technical Report 90-004, Computer Science Department, Oregon Graduate Institute (1994)

    Google Scholar 

  3. Cottrell, M., Fort, J.C., Pagès, G.: Theoretical aspects of the SOM algorithm. Neurocomputing 21(1), 119–138 (1998)

    Article  MATH  Google Scholar 

  4. Dersch, D., Tavan, P.: Asymptotic level density in topological feature maps. IEEE Trans. Neural Networks 6(1), 230–236 (1995)

    Article  Google Scholar 

  5. Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, West Sussex (2003)

    Book  MATH  Google Scholar 

  6. Fort, J.C., Cottrell, M., Letremy, P.: Stochastic on-line algorithm vs. batch algorithm for quantization and self-organizing maps. In: Neural Networks for Signal Processing XI: Proc. of the 2001 IEEE Signal Processing Society Workshop, pp. 43–52. IEEE, Piscataway (2001)

    Google Scholar 

  7. Fukunaga, K., Olsen, D.R.: An algorithm for finding intrinsic dimensionality of data. IEEE Trans. Computers C-20, 176–183 (1971)

    Article  MATH  Google Scholar 

  8. Gersho, A.: On the structure of vector quantizers. IEEE Trans. Inform. Theory IT-25, 373–380 (1979)

    Article  MathSciNet  Google Scholar 

  9. Kohonen, T.: Self-Organizing Maps, 3rd edn. Springer, Heidelberg (2001)

    Book  MATH  Google Scholar 

  10. Lewis, D.D., Yang, Y., Rose, T.G., Li, T.: RCV1: A new benchmark collection for text categorization research. J. Mach. Learn. Res. 5, 361–397 (2004)

    Google Scholar 

  11. Linde, Y., Buzo, A., Gray, R.M.: An algorithm for vector quantization. IEEE Trans. Communication COM-28, 84–95 (1980)

    Article  Google Scholar 

  12. Manning, C.D., Schütze, H.: Foundations of Statistical Natural Language Processing. MIT Press, Cambridge (1999)

    MATH  Google Scholar 

  13. McAuliffe, J.D., Atlas, L.E., Rivera, C.: A comparison of the LBG algorithm and Kohonen neural network paradigm for image vector quantization. In: Proc. ICASSP-90, Acoustics, Speech and Signal Processing, vol. IV, pp. 2293–2296. IEEE Service Center, Piscataway (1990)

    Google Scholar 

  14. Ritter, H.: Asymptotic level density for a class of vector quantization processes. IEEE Trans. Neural Networks 2(1), 173–175 (1991)

    Article  MathSciNet  Google Scholar 

  15. Vesanto, J., Alhoniemi, E., Himberg, J., Kiviluoto, K., Parviainen, J.: Self-organizing map for data mining in Matlab: the SOM Toolbox. Simulation News Europe (25), 54 (1999)

    Google Scholar 

  16. Zador, P.L.: Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory IT-28, 139–149 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Kohonen, T., Nieminen, I.T., Honkela, T. (2009). On the Quantization Error in SOM vs. VQ: A Critical and Systematic Study. In: Príncipe, J.C., Miikkulainen, R. (eds) Advances in Self-Organizing Maps. WSOM 2009. Lecture Notes in Computer Science, vol 5629. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02397-2_16

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  • DOI: https://doi.org/10.1007/978-3-642-02397-2_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02396-5

  • Online ISBN: 978-3-642-02397-2

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