Abstract
This letter addresses the asymptotic convergence of Kohonen's LVQ1 algorithm when the number of training samples are finite with an analysis that uses the dynamical systems and optimisation theories. It establishes the sufficient conditions to ensure the convergence of LVQ1 near a minimum of its cost function for constant step sizes and cyclic sampling. It also proposes a batch version of LVQ1 based on the very fast Newton optimisation method that cancels the dependence of the on-line version on the order of supplied training samples.
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References
Kohonen, T.: Self-Organizing Maps, 2nd Edn, Springer-Verlag, Berlin (1996).
LaVigna, A.: Nonparametric classification using LVQ. Ph.D. Dissertation. University of Maryland (1989).
Zhu, C. et al.: Analysis of LVQ algorithms for Pattern Classification. Proc. of ICASSP 95, 5 (1995), 3471–3474.
Bottou, L.: Online Learning and Stochastic Approximations, In D. Saal (ed.), Online Learning and Neural Networks, Cambridge University Press, Cambridge (1998).
Benveniste, A., Métivier, M. and Priouret, P.: Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, Berlin (1990).
Devaney, R. L.: An Introduction to Chaotic Dynamical Systems, 2nd edn. Reading, Addison-Wesley, Boston, MA (1989).
Hiriart-Urruty, J. B. and Lemaréchal, C.: Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin (1993).
Dennis Jr. J. E. and Schnabel, R. B.: A View of Unconstrained Optimization, In G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd (eds.), Optimization, North-Holland, Amsterdam (1989).
Hestenes, M.: Conjugate Direction Methods in Optimization, Springer-Verlag, Berlin (1980).
Kohonen, T., Hynninen, J., Kangas, J., Laaksonen, J. and Torkkola, K. Kari,: LVQPAK. The Learning Vector Quantization Program Package. Version 3.1, Laboratory of Computer and Information Science, Helsinki University of Technology (1995).
Bishop, C. M.: Neural Networks and Pattern Recognition, Oxford University Press, Oxford (1995).
Devroye, L., Györfi, L. and Lugosi, G.: A Probabilistic Theory of Pattern Recognition, Springer-Verlag, Berlin (1996).
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Bermejo, S., Cabestany, J. Finite-Sample Convergence Properties of the LVQ1 Algorithm and the Batch LVQ1 Algorithm. Neural Processing Letters 13, 135–157 (2001). https://doi.org/10.1023/A:1011328322315
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DOI: https://doi.org/10.1023/A:1011328322315