Recursive Self-organizing Map as a Contractive Iterative Function System

Recently, there has been a considerable research activity in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, the representational capabilities and internal representations of the models are not well understood. We rigorously analyze a generalization of the Self-Organizing Map (SOM) for processing sequential data, Recursive SOM (RecSOM [1]), as a non-autonomous dynamical system consisting of a set of fixed input maps. We show that contractive fixed input maps are likely to produce Markovian organizations of receptive fields on the RecSOM map. We derive bounds on parameter β (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed input maps is guaranteed.

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