c-means clustering with the ll and l∞ norms

An extension of the hard and fuzzy c-means (HCM/FCM) clustering algorithms is described. Specifically, these models are extended to admit the case where the (dis)similarity measure on pairs of numerical vectors includes two members of the Minkowski or p-norm family, viz., the p=1 and p=∞ norms. In the absence of theoretically necessary conditions to guide a numerical solution of the nonlinear constrained optimization problem associated with this case, it is shown that a certain basis exchange algorithm can be used to find approximate critical points of the new objective functions. This method broadens the applications horizon of the FCM family by enabling users to match discontinuous multidimensional numerical data structures with similarity measures that have nonhyperelliptical topologies