Consistency and Inconsistency in $K$-Means Clustering
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918086525845504 |
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| author | Blanchard, Moïse Jaffe, Adam Quinn Zhivotovskiy, Nikita |
| author_facet | Blanchard, Moïse Jaffe, Adam Quinn Zhivotovskiy, Nikita |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Consistency and Inconsistency in $K$-Means Clustering Blanchard, Moïse Jaffe, Adam Quinn Zhivotovskiy, Nikita Statistics Theory Probability Machine Learning 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. |
| title | Consistency and Inconsistency in $K$-Means Clustering |
| topic | Statistics Theory Probability Machine Learning |
| url | https://arxiv.org/abs/2507.06226 |