Analyzing phase transitions in high-dimensional self-organizing maps

Abstract.

The self-organizing map (SOM), a widely used algorithm for the unsupervised learning of neural maps, can be formulated in a low-dimensional ‘feature map’ variant which requires prespecified parameters (‘features’) for the description of receptive fields, or in a more general high-dimensional variant which allows self-organization of the structure of individual receptive fields as well as their arrangement in a map. We present here a new analytical method for deriving conditions for the emergence of structure in SOMs which is particularly suited for the as yet inaccessible high-dimensional SOM variant. Our approach is based on an evaluation of a map distortion function. It involves only an ansatz for the way stimuli are distributed among map neurons; the receptive fields of the map need not be known explicitly. Using this method we first calculate regions of stability for four possible states of SOMs projecting from a rectangular input space to a ring of neurons. We then analyze the transition from nonoriented to oriented receptive fields in a SOM-based model for the development of orientation maps. In both cases, the analytical results are well corroborated by the results of computer simulations.