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.