$L_{p}$-norm probabilistic K-means clustering via nonlinear programming

Abstract

Generalized fuzzy c-means (GFCM) is an extension of fuzzy c-means using $L_{p}$-norm distances. However, existing methods cannot solve GFCM with m = 1. To solve this problem, we define a new kind of clustering models, called $L_{p}$-norm probabilistic K-means ($L_{p}$-PKM). Theoretically, $L_{p}$-PKM is equivalent to GFCM at m = 1, and can have nonlinear programming solutions based on an efficient active gradient projection (AGP) method, namely, inverse recursion maximum-step active gradient projection (IRMSAGP). On synthetic and UCI datasets, experimental results show that $L_{p}$-PKM performs better than GFCM (m > 1) in terms of initialization robustness, p-influence, and clustering performance, and the proposed IRMSAGP also achieves better performance than the traditional AGP in terms of convergence speed.