Chromatic Clustering in High Dimensional Space

In this paper, we study a new type of clustering problem, called {\em Chromatic Clustering}, in high dimensional space. Chromatic clustering seeks to partition a set of colored points into groups (or clusters) so that no group contains points with the same color and a certain objective function is optimized. In this paper, we consider two variants of the problem, chromatic $k$-means clustering (denoted as $k$-CMeans) and chromatic $k$-medians clustering (denoted as $k$-CMedians), and investigate their hardness and approximation solutions. For $k$-CMeans, we show that the additional coloring constraint destroys several key properties (such as the locality property) used in existing $k$-means techniques (for ordinary points), and significantly complicates the problem. There is no FPTAS for the chromatic clustering problem, even if $k=2$. To overcome the additional difficulty, we develop a standalone result, called {\em Simplex Lemma}, which enables us to efficiently approximate the mean point of an unknown point set through a fixed dimensional simplex. A nice feature of the simplex is its independence with the dimensionality of the original space, and thus can be used for problems in very high dimensional space. With the simplex lemma, together with several random sampling techniques, we show that a $(1+\epsilon)$-approximation of $k$-CMeans can be achieved in near linear time through a sphere peeling algorithm. For $k$-CMedians, we show that a similar sphere peeling algorithm exists for achieving constant approximation solutions.