An Affine-Invariant Bayesian Cluster Process

Abstract: In order to identify clusters of objects with features transformed by unknown affine transformations, we develop a Bayesian cluster process which is invariant with respect to certain linear transformations of the feature space and able to cluster data without knowing the number of clusters in advance. Specifically, our proposed method can identify clusters invariant to orthogonal transformations under model I, invariant to scaling-coordinate orthogonal transformations under model II, or invariant to arbitrary non-singular linear transformations under model III. The proposed split-merge algorithm leads to an irreducible and aperiodic Markov chain, which is also efficient at identifying clusters reasonably well for various applications. We illustrate the applications of our approach to both synthetic and real data such as leukemia gene expression data for model I; wine data and two half-moons benchmark data for model II; three-dimensional Denmark road geographic coordinate system data and an arbitrary non-singular transformed two half-moons data for model III. These examples show that the proposed method could be widely applied in many fields, especially for finding the number of clusters and identifying clusters of samples of interest in aerial photography and genomic data.