  The $k$-means is one of the most important unsupervised learning techniques
in statistics and computer science. The goal is to partition a data set into
many clusters, such that observations within clusters are the most homogeneous
and observations between clusters are the most heterogeneous. Although it is
well known, the investigation of the asymptotic properties is far behind,
leading to difficulties in developing more precise $k$-means methods in
practice. To address this issue, a new concept called clustering consistency is
proposed. Fundamentally, the proposed clustering consistency is more
appropriate than the previous criterion consistency for the clustering methods.
Using this concept, a new $k$-means method is proposed. It is found that the
proposed $k$-means method has lower clustering error rates and is more robust
to small clusters and outliers than existing $k$-means methods. When $k$ is
unknown, using the Gap statistics, the proposed method can also identify the
number of clusters. This is rarely achieved by existing $k$-means methods
adopted by many software packages.
