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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:

Seo, S. and Obermayer, K.
(2004). Self-Organizing Maps and Clustering Methods for Matrix Data.
*Neural Networks Special Issue*, 17, 1211 – 1229.

Seo, S., Bode, M. and Obermayer, K.
(2003). Soft Nearest Prototype Classification.
*IEEE Transactions on Neural Networks*, 14, 390 – 398.

Seo, S. and Obermayer, K.
(2003). Soft Learning Vector Quantization.
*Neural Computation*, 15, 1589 – 1604.

Graepel, T. and Obermayer, K.
(1999). A Self-Organizing Map for Proximity Data.
*Neural Computation*, 11, 139 – 155.

Hasenjäger, M., Ritter, H. and Obermayer, K. (1999). Active Learning in Self-Organizing Maps. *Kohonen Maps*. Elsevier, 57–70.,

Graepel, T., Burger, M. and Obermayer, K.
(1998). Self-Organizing Maps: Generalizations and New Optimization Techniques.
*Neurocomputing*, 20, 173 – 190.

Graepel, T., Burger, M. and Obermayer, K.
(1997). Phase Transitions in Stochastic Self-Organizing Maps.
*PHYSICAL REVIEW E*, 56, 3876 – 3890.

Erwin, E., Obermayer, K. and Schulten, K.
(1992). Self-Organizing Maps: Ordering, Convergence Properties and Energy Functions.
*Biological Cybernetics*, 67, 47 – 55.

Erwin, E., Obermayer, K. and Schulten, K.
(1992). Self-Organizing Maps: Stationary States, Metastability and Convergence Rate.
*Biological Cybernetics*, 67, 35 – 45.