Pinch Ratio Clustering from a Topologically Intrinsic Lexicographic Ordering

This paper introduces an algorithm for determining data clusters called TILO/PRC (Topologically Intrinsic Lexicographic Ordering/Pinch Ratio Clustering). The theoretical foundation for this algorithm, developed in [14], uses ideas from topology (particularly knot theory) suggesting that it should be very flexible and robust with respect to noise. The TILO portion of the algorithm progressively improves a linear ordering of the points in a data set until the ordering satisfies a topological condition called strongly irreducible. The PRC algorithm then divides the data set based on this ordering and a heuristic metric called the pinch ratio. We demonstrate the effectiveness of TILO/PRC for finding clusters in a wide variety of real and synthetic data sets and compare the results to existing clustering methods. Moreover, because the output of TILO depends on the initial ordering, we consider the effects of different random orderings on the final clusters defined by PRC, and show that choosing an initial ordering based on a different clustering algorithm can improve the final clusters. These results verify that both the theoretical foundations of TILO and the heuristic notion of pinch ratio are reasonable.