  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.
