The magnification exponents ¯ occuring in adaptive map formation algorithms like Kohonen's self-organizing feature map deviate for the information theoretically optimal value ¯ = 1 as well as from the values which optimize, e.g., the mean square distortion error (¯ = 1=3 for one-dimensional maps). At the same time, models for categorical perception such as the "perceptual magnet" effect which are based on topographic maps require negative magnification exponents ¯ ! 0. We present an extension of the self-organizing feature map algorithm which utilizes adaptive local learning step sizes to actually control the magnification properties of the map. By change of a single parameter, maps with optimal information transfer, with various minimal reconstruction errors, or with an inverted magnification can be generated. Analytic results on this new algorithm are complemented by numerical simulations. 1. Introduction The representation of information in topographic maps is a common property of...

Self-organizing feature maps with self-determined local neighborhood widths are applied to construct principal manifolds of data distributions. This task exemplifies the problem of the learning of learning parameters in neural networks. The proposed algorithm is based upon analytical results on phase transitions in self-organizing feature maps available for idealized situations. By illustrative simulations it is demonstrated that deviations from the theoretically studied situation are compensated adaptively and that the capability of topology preservation is crucial for avoiding overfitting effects. Further, the relevance of the parameter learning scheme for hierarchical feature maps is stated. 1 Introduction Many learning algorithms are influenced by the choice and by the course of on-line modification of particular parameters. Whereas the learning algorithm itself represents a formalized principle such as the minimization of an error measure, principles to govern parameter settings ...

Introduction Kohonen's self-organizing feature maps (SOFMs) [8] usually exhibit a selective magnification of often stimulated regions of their input space. This amounts to a larger transmission of information about the stimulus ensemble than in maps with a constant resolution. Such a selective magnification is not only observed in biological maps, but is also often regarded as a desirable design objective in technical contexts. For at least three reasons, the magnification properties of SOFMs deserve further investigation: 1. An analysis by Ritter and Schulten [10] demonstrated that the SOFM algorithm does not yield a maximum entropy map (i.e. does not transmit the maximum amount of information). 2. As a related argument we observe that it depends on the error criterion one applies which magnification properties are to be regarded as optimal. For example, a minimal worst case error is achieved by maps with all receptive fields (or Voronoy polygons) being of equal extension, i.