Strong Consistency of Reduced K ‐means Clustering

Reduced k ‐means clustering is a method for clustering objects in a low‐dimensional subspace. The advantage of this method is that both clustering of objects and low‐dimensional subspace reflecting the cluster structure are simultaneously obtained. In this paper, the relationship between conventional k ‐means clustering and reduced k ‐means clustering is discussed. Conditions ensuring almost sure convergence of the estimator of reduced k ‐means clustering as unboundedly increasing sample size have been presented. The results for a more general model considering conventional k ‐means clustering and reduced k ‐means clustering are provided in this paper. Moreover, a consistent selection of the numbers of clusters and dimensions is described.