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- T. Graepel and
K. Obermayer. A Self-Organizing Map for Proximity Data.
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Neural Comput., 11:139-155, 1999.
(FTP Gzipped PostScript, 17 pages, 87 kb)
We derive an efficient algorithm for topographic mapping of
proximity data (TMP), which can be seen as an extension of Kohonens
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 mathcal
D and the topographic neighborhood via a matrix mathcal 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 mathcal D and mathcal 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
cats cerebral cortex.
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