Partial k-means to avoid outliers, mathematical programming formulations, complexity results

Nicolas Dupin; Frank Nielsen

doi:10.1007/978-3-031-34020-8_22

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arXiv · 2302.05644

Partial k-means to avoid outliers, mathematical programming formulations, complexity results

Nicolas Dupin·Frank Nielsen

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Abstract

A well-known bottleneck of Min-Sum-of-Square Clustering (MSSC, the celebrated $k$-means problem) is to tackle the presence of outliers. In this paper, we propose a Partial clustering variant termed PMSSC which considers a fixed number of outliers to remove. We solve PMSSC by Integer Programming formulations and complexity results extending the ones from MSSC are studied. PMSSC is NP-hard in Euclidean space when the dimension or the number of clusters is greater than $2$. Finally, one-dimensional cases are studied: Unweighted PMSSC is polynomial in that case and solved with a dynamic programming algorithm, extending the optimality property of MSSC with interval clustering. This result holds also for unweighted $k$-medoids with outliers. A weaker optimality property holds for weighted PMSSC, but NP-hardness or not remains an open question in dimension one.

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BibTeX RIS

Nicolas Dupin, Frank Nielsen. 2023-02-11. Partial k-means to avoid outliers, mathematical programming formulations, complexity results. https://doi.org/10.1007/978-3-031-34020-8_22

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