The classical $k$-means clustering requires a complete data matrix without missing entries. As a natural extension of the $k$-means clustering for missing data, the $k$-POD clustering has been proposed, which ignores the missing entries in the $k$-means clustering. This paper shows the inconsistency of the $k$-POD clustering even under the missing completely at random mechanism. More specifically, the expected loss of the $k$-POD clustering can be represented as the weighted sum of the expected $k$-means losses with parts of variables. Thus, the $k$-POD clustering converges to the different clustering from the $k$-means clustering as the sample size goes to infinity. This result indicates that although the $k$-means clustering works well, the $k$-POD clustering may fail to capture the hidden cluster structure. On the other hand, for high-dimensional data, the $k$-POD clustering could be a suitable choice when the missing rate in each variable is low.

翻译：经典的$k$-均值聚类要求数据矩阵完整无缺失项。作为缺失数据$k$-均值聚类的自然扩展，$k$-POD聚类被提出，该方法在$k$-均值聚类过程中忽略缺失项。本文证明了即使在完全随机缺失机制下，$k$-POD聚类也存在不一致性。具体而言，$k$-POD聚类的期望损失可表示为部分变量上期望$k$-均值损失的加权和。因此，当样本量趋于无穷时，$k$-POD聚类会收敛到与$k$-均值聚类不同的划分结果。这一结果表明，即使$k$-均值聚类表现良好，$k$-POD聚类仍可能无法捕捉潜在的簇结构。另一方面，对于高维数据，当各变量缺失率较低时，$k$-POD聚类可能是一个合适的选择。