Minkowski Metric Based Soft Subspace Clustering with Different Minkowski Exponent and Feature Weight Exponent

Abstract

Soft subspace clustering (SSC) methods can simultaneously performance clustering and find the subspace where each cluster lie in. A Minkowski metric based SSC (MSSC) algorithm recently is proposed to improve the adaptability of SSC to data. The empirical results have shown its favorable performance in comparison with several other popular clustering algorithms. However, this algorithm has the following two main defects: (1) The role that the Minkowski exponent $\beta$ in MSSC plays is not clear. And the Minkowski exponent $\beta$ is set as the same as the feature weight exponent $\alpha$ in MSSC that may lead to MSSC missing better clustering performance. (2) the steepest descent method based MSSC (SD-MSSC) is computationally expensive for large data. In this paper, a general formulation for MSSC is presented, in which the Minkowski exponent $\beta$ can be set not equal to the feature weight exponent $\alpha$. A novel algorithm for computing the clustering centroids in MSSC is presented using the fixed-point iteration (FPI) method. The FPI based MSSC (FPI-MSSC) algorithm is more efficient than the SD-MSSC algorithm, and it becomes a noise-robust SSC procedure when the $\beta <2$. Extensive experiments on real-world data sets are presented to show the effectiveness of the proposed algorithm.