Phase Transitions in Self-Organized Feature Maps

Abstract

We present an analytical study of symmetry-breaking phenomena in Kohonen's self-orga­ ruzed feature map (SOFM) that occur due to a topological mismatch between the input space and the neuron setup. We give a microscopic derivation for the time dependent Ginzburg­ Landau equations describing the behavior of the order parameter close to the critical point where the phase transition takes place. The Kohonen algorithm [T] provides a simple model of the self-organized generation of an internal representation of the environment in living systems. [T] For practical applications it may be used as a nonlinear generalization of principal component analysis which plays an important role in many problems of data analysis, above all for data in high dimensional data spaces. Considering from the point of view of physics the synaptic vectors as coordinates of a set of hypothetical particles with a complex dynamics created by the Kohonen algorithm. The complexity of this system is reflected by phenomena like spontaneous symmetry breaking (analogous to noneqnilibrium phase transitions) and the occurrence of metastable states with partial ordering or the emergence of criticality under certain conditions [T]. From the point of view of systems theory Kohonen's algorithm models a system with competition and cooperation which may serve as a generic though degenerate example of the emergence of collective ordering phenomena in more complicated systems. Concerning practical applications these effects may well interfere with the intentions of using the feature maps as a reliable tool for topology preserving mapping. On the other hand, if better understood they might be exploited for more sophisticated applications. In this sense, there is a great lack of theoretical study into the details of the map evolution. Pioneering work on this task has been performed by Ritter, Martinetz & Schulten [T]. They predicted a phase transition to occur due to dimensional conflicts between input space and net topology and calculated within the linear theory the critical value for the corresponding control parameter which measures the strength of the dimensional conflict. In the present paper we present by the means of the time dependent Ginzburg-Landau formalism a general approach for the evaluation of order parameters in the SOFM. In particular, by including nonlinear terms we describe the behavior around the critical point where the phase transition takes place. The critical index and a modulating prefactor were obtained. The Ginzburg-Landau approach is quite general but we study here the simple case of mapping a two-dimensional input space onto a one-dimensional neuron chain. In input space the data points are assumed to be distributed homogeneously in a box of height s which plays the role of the control parameter measuring the strength of the dimensional mismatch.