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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 | Graepel1999b |
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Author | Graepel, T. and Obermayer, K. |

Pages | 139 – 155 |

Year | 1999 |

Journal | Neural Computation |

Volume | 11 |

Abstract | We derive an efficient algorithm for topographic mapping of proximity data (TMP), which can be seen as an extension of Kohonen\'s Self-Organizing Map to arbitrary distance measures. The TMP cost function is derived in a Baysian framework of Folded Markov Chains for the description of autoencoders. It incorporates the data via a dissimilarity matrix $D$ and the topographic neighborhood via a matrix $H$ of transition probabilities. From the principle of Maximum Entropy a non-factorizing Gibbs-distribution is obtained, which is approximated in a mean-field fashion. This allows for Maximum Likelihood estimation using an EM algorithm. In analogy to the transition from Topographic Vector Quantization (TVQ) to the Self-organizing Map (SOM) we suggest an approximation to TMP which is computationally more efficient. In order to prevent convergence to local minima, an annealing scheme in the temperature parameter is introduced, for which the critical temperature of the first phase-transition is calculated in terms of $D$ and $H$. Numerical results demonstrate the working of the algorithm and confirm the analytical results. Finally, the algorithm is used to generate a connection map of areas of the cat\'s cerebral cortex. |

Bibtex Type of Publication | Selected:quantization |