Strong Consistency of Sparse K-means Clustering

In this paper, we prove the strong consistency of the sparse K-means method proposed by Witten and Tibshirani (2010). We prove the consistency in both risk and clustering for the Euclidean distance. We discuss the characterization of the limit of the clustering under some special cases. For the general distance, we prove the consistency in risk. Our result naturally extends to other models with the same objective function but different constraints such as l0 or l1 penalty in Chang et al. (2018).