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Phase Transitions in Soft Topographic Vector Quantization
Citation key Burger1997
Author Burger, M. and Graepel, T. and Obermayer, K.
Title of Book Artificial Neural Networks - ICANN 97
Pages 619 – 624
Year 1997
Editor W. Gerstner and A. Germond and M. Hasler and J. Nicoud
Publisher Springer-Verlag
Abstract We have developed an algorithm (STVQ) for the optimization of neighbourhood preserving maps by applying deterministic annealing to an energy function for topographic vector quantization. The combinatorial optimization problem is solved by introducing temperature dependent fuzzy assignments of data points to cluster centers and applying an EM-type algorithm at each temperature while annealing. The annealing process exhibits phase transitions in the cluster representation for which we calcul ate critical modes and temperatures expressed in terms of the neighbourhood function and the covariance matrix of the data. In particular, phase transitions corresponding to the automatic selection of feature dimensions are explored analytically and numer ically for finite temperatures. Results are related to those obtained earlier for Kohonen\'s SOM-algorithm which can be derived as an approximation to STVQ. The deterministic annealing approach makes it possible to use the neighbourhood function solely to encode desired neighbourhood relations. The working of the annealing process is visualized by showing the effects of ``heating\'\' on the topological structure of a two-dimensional map of the plane.
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