Some notes on the $k$-means clustering for missing data

TL;DR

This paper critically examines the $k$-POD clustering method for missing data, revealing its inconsistency and potential failure to identify true clusters, especially in large samples, while noting its suitability for high-dimensional, low-missing-rate data.

Contribution

The paper demonstrates the inconsistency of $k$-POD clustering under missing completely at random and clarifies its limitations compared to traditional $k$-means clustering.

Findings

$k$-POD clustering converges to a different clustering than $k$-means as sample size increases.

$k$-POD may fail to capture true cluster structures in large samples.

$k$-POD can be effective in high-dimensional data with low missing rates.

Abstract

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.