Feature Space Curvature Map: A Method To Homogenize Cluster Densities

Abstract

The majority of density-based clustering algorithms can not perform properly when data expose very different density through the feature space. These algorithms implicitly presume that all clusters almost have the same density, therefore, they normally use global parameters. Consequently, they are often biased towards finding dense clusters in front of sparse ones. In this paper, we propose a parametric multilinear transformation method to homogenize cluster densities while preserving the topological structure of the dataset. The transformed clusters have approximately the same density while all inter-cluster regions become globally low-density. In our method, the feature space is locally bent by dense data point concentrations the same way as stars bend the space-time dimensions in Theory of Relativity. We present a new Gravitational Self-organization Map to model the feature space curvature by plugging the concepts of gravity and fabric of space into the Self-organization Map algorithm to mathematically describe the density structure of the data. To homogenize the cluster density, we introduce a novel mapping mechanism to project the data from a non-Euclidean curved space to a new Euclidean flat space. Specifically, this mechanism transfers the basis vectors instead of the feature vectors to guarantee the continuity of the mapping function and optimize the computation cost of the algorithm. As a result, our method can efficiently and explicitly homogenize the density of any dataset globally to then apply existing clustering algorithms without modification. Our experimental results over both realworld and synthetic datasets show that our approach outperforms the current statistical-based methods.