Dempster-Shafer clustering using Potts spin mean field theory

In this article we investigate a problem within Dempster±Shafer theory where 2 q À 1 pieces of evidence are clustered into q clusters by minimizing a metacon¯ict function, or equivalently, by minimizing the sum of weight of con¯ict over all clusters. Previously one of us developed a method based on a Hop®eld and Tank model. However, for very large problems we need a method with lower computational complexity. We demonstrate that the weight of con¯ict of evidence can, as an approximation, be linearized and mapped to an antiferromagnetic Potts spin model. This facilitates ef®cient numerical solution, even for large problem sizes. Optimal or nearly optimal solutions are found for Dempster±Shafer clustering benchmark tests with a time complexity of approximately O…N 2 log 2 N†. Furthermore, an isomorphism between the antiferromagnetic Potts spin model and a graph optimization problem is shown. The graph model has dynamic variables living on the links, which have a priori probabilities that are directly related to the pairwise con¯ict between pieces of evidence. Hence, the relations between three different models are shown. 1 Introduction In this article we develop a method for clustering evidence in very large scale problems within Dempster±Shafer theory [14, 15, 40±44, 46]. We consider the case when evidence come from multiple events which should be handled independently, and it is not known to which event a piece of evidence is related. We use the clustering process to separate the evidence into subsets for each event, so that each subset may be handled separately. In an earlier article [38] one of us developed a method using a neural network structure similar to the Hop®eld and Tank model [22] for partitioning evidence into clusters for relative large scale problems. All the weights were set a priori using the con¯ict in Dempster's rule, thus no learning process was utilized. This clustering approach represented a great improvement in computational complexity compared to a previous method based on iterative optimization [29±35], although its clustering performance was not equally good. In order to improve clustering performance a hybrid of the two methods was also developed [36]. In a recent paper [37] this method was further extended for simultaneous clustering and determination of number of clusters during iteration in the neural structure. Here, we let the neuron output signals represent the degree to which pieces of evidence belong to corresponding clusters. From these signals we derive a probability distribution …