L2,p-Norm Based Discriminant Subspace Clustering Algorithm

Abstract

Discriminative subspace clustering (DSC) combines Linear Discriminant Analysis (LDA) with clustering algorithm, such as K-means (KM), to form a single framework to perform dimension reduction and clustering simultaneously. It has been verified to be effective for high-dimensional data. However, most existing DSC algorithms rigidly use the Frobenius norm (F-norm) to define model that may not always suitable for the given data. In this paper, DSC is extended in the sense of $l_{2, p}$ -norm, which is a general form of the F-norm, to obtain a family of DSC algorithms which provide more alternative models for practical applications. In order to achieve this goal. Firstly, an efficient algorithm for the $l_{p}$ -norm based KM (KM $_{\mathrm {p}}$ ) clustering is proposed. Then, based on the equivalence of LDA and linear regression, a $l_{\mathrm {2,}p}$ -norm based LDA ( $l_{\mathrm {2,}p}$ -LDA) is proposed, and an efficient Iteratively Reweighted Least Squares algorithm for $l_{\mathrm {2,}p}$ -LDA is presented. Finally, KMp and $l_{2, p}$ -LDA are combined into a single framework to form an efficient generalized DSC algorithm: $l_{2,{p}}$ -norm based DSC clustering ( $l_{2,{p}}$ -DSC). In addition, the effects of the parameters on the proposed algorithm are analyzed, and based on the theory of robust statistics, a special case of $l_{2,{p}}$ -DSC, which can show better robustness on the data sets with noise and outlier, is studied. Extensive experiments are performed to verify the effectiveness of our proposed algorithm.