Nonparametric density estimation and regression achieved with topographic maps maximizing the information-theoretic entropy of their outputs

We introduce an unsupervised competitive learning rule, called the extended Maximum Entropy learning Rule (eMER), for topographic map formation. Unlike Kohonen's Self-Organizing Map (SOM) algorithm, the presence of a neighborhood function is not a prerequisite for achieving topology-preserving mappings, but instead it is intended: (1) to speed up the learning process and (2) to perform nonparametric regression. We show that, when the neighborhood function vanishes, the neural weigh t density at convergence approaches a linear function of the input density so that the map can be regarded as a nonparametric model of the input density. We apply eMER to density estimation and compare its performance with that of the SOM algorithm and the variable kernel method. Finally, we apply the ‘batch’ version of eMER to nonparametric projection pursuit regression and compare its performance with that of back-propagation learning, projection pursuit learning, constrained topolog ical mapping, and the Heskes and Kappen approach. Received: 12 August 1996 / Accepted in revised form: 9 April 1997