SPARSE k-MEANS WITH ℓ∞/ℓ0 PENALTY FOR HIGH-DIMENSIONAL DATA CLUSTERING - Chang, Xiangyu; Wang, Yu; Li, Rongjian; Xu, Zongben (Statistica Sinica, 2018)
One of the existing sparse clustering approaches, ℓ1-k-means, maximizes the weighted between-cluster sum of squares subject to the ℓ1 penalty. In this paper, we propose a sparse clustering method based on an ℓ∞/ℓ0 penalty, which we call ℓ0-k-means. We design an efficient iterative algorithm for solving it. To compare the theoretical properties of ℓ1 and ℓ0-k-means, we show that they can be explained explicitly from a thresholding perspective based on different thresholding functions. Moreover, ℓ1 and ℓ0-k-means are proven to have a screening consistent property under Gaussian mixture models. Experiments on synthetic as well as real data justify the outperforming results of ℓ0 with respect to ℓ1-k-means.