Consistency and Inconsistency in $K$-Means Clustering

• URL: http://arxiv.org/abs/2507.06226v1
• Date: Tue, 08 Jul 2025 17:59:02 GMT
• Title: Consistency and Inconsistency in $K$-Means Clustering
• Authors: Moïse Blanchard, Adam Quinn Jaffe, Nikita Zhivotovskiy,
• Abstract summary: A celebrated result of Pollard proves consistency for $k$-means clustering when the population distribution has finite variance.<n>We show that the population-level $k$-means clustering problem is, in fact, well-posed under the weaker assumption of a finite expectation.
• Score: 10.234307333602048
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A celebrated result of Pollard proves asymptotic consistency for $k$-means clustering when the population distribution has finite variance. In this work, we point out that the population-level $k$-means clustering problem is, in fact, well-posed under the weaker assumption of a finite expectation, and we investigate whether some form of asymptotic consistency holds in this setting. As we illustrate in a variety of negative results, the complete story is quite subtle; for example, the empirical $k$-means cluster centers may fail to converge even if there exists a unique set of population $k$-means cluster centers. A detailed analysis of our negative results reveals that inconsistency arises because of an extreme form of cluster imbalance, whereby the presence of outlying samples leads to some empirical $k$-means clusters possessing very few points. We then give a collection of positive results which show that some forms of asymptotic consistency, under only the assumption of finite expectation, may be recovered by imposing some a priori degree of balance among the empirical $k$-means clusters.

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