In the present paper, we address the problem of extrapolating group proximities from member relations, which we refer to as the group proximity problem. We assume that a relational dataset consists of several groups and that pairwise relations of all members can be measured. Under these assumptions, the goal is to estimate group proximities from pairwise relations. In order to solve the group proximity problem, we present a method based on embedding and distribution mapping, in which all relational data, which consist of pairwise dissimilarities or dissimilarities between members, are transformed into vectorial data by embedding methods. After this process, the distributions of the groups are obtained. Group proximities are estimated as distances between distributions by distribution mapping methods, which generate a map of distributions. As an example, we apply the proposed method to document and bacterial flora datasets. Finally, we confirm the feasibility of using the proposed method to solve the group proximity problem.

This paper presents an efficient algorithm for large-scale multi-system learning task. The proposed architecture, referred to as the 'RBF×SOM', is based on the SOM2, that is, a'SOM of SOMs'. As is the case in the modular network SOM (mnSOM) with multilayer perceptron modules (MLP-mnSOM), the aim of the RBF×SOM is to organize a continuous map of nonlinear functions representing multi-class input-output relations of the given datasets. By adopting the algorithm for the SOM2, the RBF×SOM generates a map much faster than the original mnSOM, and without the local minima problem. In addition, the RBF×SOM can be applied to more difficult cases, that were not easily dealt with by the MLP-mnSOM. Thus, the RBF×SOM can deal with cases in which the probability density of the inputs is dependent on the classes. This tends to happen more often as the input dimension increases. The RBF×SOM therefore, overcomes many of the problems inherent in the MLP-mnSOM, and this is crucial for application to large scale tasks. Simulation results with artificial datasets and a meteorological dataset confirm the performance of the RBF×SOM.