Trajectory decomposition for unsupervised learning neural models

In this paper, use the trajectory decomposition technique to rigorously prove the similarity between different neural models. First, we derive the free energy function for an unsupervised net of stochastic neurons with lateral interactions. The temperature incorporated in this function serves as control parameter in the annealing schedule. Then, we consider the incremental and batch modes of learning resulting in corresponding versions of soft topologypreserving mapping. The mapping utilizes only weak lateral interactions that can be, therefore, approximated by the nearest-neighbour ones. Considering the weight vector of a neuron as a “particle” moving in the spacetime of imposed patterns, we decompose this particle trajectory over these patterns. Using the decomposition for incremental and batch modes of soft topology-preserving map, we derive the cortical map and the elastic net respectively [1,2]. The temperature of the above maps is transformed into the Gaussian variance of the cortical map and the elastic net. This fact elucidates indirect incorporation of soft competition and deterministic annealing into the cortical map and the elastic net, which makes them to be very powerful neurocomputational models. We show that the batch version of soft topology-preserving map is rigourously reduced to the corresponding elastic net. Unlike, the incremental version of soft topology-preserving map is reduced to the cortical map only in the low temperature limit. We tested the models on the relevant to them tasks: the travelling salesman problem and the development of visual cortex topology, namely the formation of retinotopy and ocular dominance. The difference in derivation of the latter systems results into the difference in their behaviour: the batch soft topology-preserving map and the elastic net produce similar outputs whereas the incremental soft topology-preserving map and the cortical map behave differently.