Thore Graepel and Klaus Obermayer
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.
|Published in||Neural Computation|