A Scale Dependent Model for Clustering by Optimization of Homogeneity and Separation ∗

We present a model for clustering a set of high dimensional data into subsets of homogenous clusters which are well-separated from each other. A novel feature of our model is that it allows the user to directly control the scale of the clusters. This is realized by formulating the clustering problem as an optimization problem whose objective function combines two measures of the quality of a clustering, namely homogeneity and separation, and a parameter which controls the scale. As an illustration to the use of our general framework, an application on clustering data based on pair-wise similarity measured by the (uncentered) Pearson correlation coefficient (cosine of the angle between two data vectors in an Euclidean space) is presented. In this case, for a dataset of size n in a p-dimensional space, our algorithm will output a clustering in O(npd + mpd2) floating point operations while typical hierarchical and partitioning algorithms require O(n2p) and O(npkm) respectively. Here d is the number of clusters determined by our algorithm, k is the number of clusters specified by the user, and m is the number of iterations. Experimental results on synthetic, biological and textured image data are presented to demonstrate the usefulness of the proposed model.