Feature-weighted clustering with inner product induced norm based dissimilarity measures: an optimization perspective

The performance of a clustering algorithm can be improved by assigning appropriate weights to different features of the data. Such feature weighting is likely to reduce the effect of noise and irrelevant features while enhancing the effect of the discriminative features simultaneously. For the clustering purpose, feature-weighted dissimilarity measures are so far limited to Euclidean, Mahalanobis, and exponential distances. In this article, we introduce a novel feature weighting scheme for the general class of inner product induced norm (IPIN) based weighted dissimilarity measures. This class has a wide range and includes the three above-mentioned distances as special cases. We develop the general algorithms to solve the hard (k-means) and fuzzy (fuzzy c-means) partitional clustering problems and undertake in-depth analyses of the convergence of the algorithms as well. In addition, we address issues like feasibility and uniqueness of the solutions of these problems in sufficient details. The novelty of the article lies in the introduction of a general feature weighting scheme for the generalized class of IPIN-based dissimilarity measures and a complete convergence analysis for the automated feature-weighted clustering algorithms using such measures.