Clustering via Concave Minimization

Part of Advances in Neural Information Processing Systems 9 (NIPS 1996)

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Paul Bradley, Olvi Mangasarian, W. Street


The problem of assigning m points in the n-dimensional real space Rn to k clusters is formulated as that of determining k centers in Rn such that the sum of distances of each point to the nearest center is minimized. If a polyhedral distance is used, the problem can be formulated as that of minimizing a piecewise-linear concave function on a polyhedral set which is shown to be equivalent to a bilinear program: minimizing a bilinear function on a polyhe(cid:173) dral set. A fast finite k-Median Algorithm consisting of solving few linear programs in closed form leads to a stationary point of the bilinear program. Computational testing on a number of real(cid:173) world databases was carried out. On the Wisconsin Diagnostic Breast Cancer (WDBC) database, k-Median training set correct(cid:173) ness was comparable to that of the k-Mean Algorithm, however its testing set correctness was better. Additionally, on the Wisconsin Prognostic Breast Cancer (WPBC) database, distinct and clini(cid:173) cally important survival curves were extracted by the k-Median Algorithm, whereas the k-Mean Algorithm failed to obtain such distinct survival curves for the same database.