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# Learning Vector Quantization and Self-organizing Maps

Self-organizing maps, often termed Kohonen maps, are a versatile and widely used tool for exploratory data analysis. Here we were interested in mathematically characterizing the embedding properties of the Self-organizing Map. We proposed robust learning schemes using deterministic annealing and we investigated extensions of the Self-organizing Map to relational data representations which included pairwise data as a special case. Emphasis was given to formulations which are based on cost-functions and optimization, and we investigated, how the different variants of the Self-organizing map relate to each other and to the original Kohonen map. We also studied prototype-based classifiers related to Learning Vector Quantization with a particular focus on improved learning schemes. Self-organizing maps were also investigated in the context of understanding self-organization and pattern formation in neural development. For details see "Research" page "Models of Neural Development".

Acknowledgement: Research was funded by the Technische Universität Berlin.

### Selected Publications:

Citation key | Erwin1992b |
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Author | Erwin, E. and Obermayer, K. and Schulten, K. |

Pages | 47 – 55 |

Year | 1992 |

Journal | Biological Cybernetics |

Volume | 67 |

Abstract | We investigate the convergence properties of the self-organizing feature map algorithm for a simple, but very instructive case: the formation of a topographic representation of the unit interval [0, 1] by a linear chain of neurons. We extend the proofs of convergence of Kohonen and of Cottrell and Fort to hold in any case where the neighborhood function, which is used to scale the change in the weight values at each neuron, is a monotonically decreasing function of distance from the winner neuron. We prove that the learning dynamics cannot be described by a gradient descent on a single energy function, but may be described using a set of potential functions, one for each neuron, which are independently minimized following a stochastic gradient descent. We derive the correct potential functions for the one- and multi-dimensional case, and show that the energy functions given by Tolat (1990) are an approximation which is no longer valid in the case of highly disordered maps or steep neighborhood functions. |

Bibtex Type of Publication | Selected:quantization |