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- M. Burger, T. Graepel, and
K. Obermayer. Phase Transitions in Soft Topographic Vector Quantization.
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In W. Gerstner, A. Germond, M. Hasler, and J. Nicoud, editors, Artificial
Neural Networks - ICANN 97, pages 619-624. Springer-Verlag, 1997.
(FTP Gzipped PostScript, 6 pages, 71 kb)
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 Kohonens 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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