Clustering via Concave Minimization

Abstract

The problem of assigning m points in the n-dimensional real space R^n to k clusters is formulated as that of determining k centers in R^n such that the sum of distance of each point to the nearest center in 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 polyhedral 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-world databases was carried out. On the Wisconsin Diagnostic Breast Cancer (WDBC) database, k-Median training set correctness 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 clinically 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.