Partial k-means to avoid outliers, mathematical programming formulations, complexity results
Abstract
A well-known bottleneck of Min-Sum-of-Square Clustering (MSSC, the celebrated -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 . 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 -medoids with outliers. A weaker optimality property holds for weighted PMSSC, but NP-hardness or not remains an open question in dimension one.
Keywords
Cite
@article{arxiv.2302.05644,
title = {Partial k-means to avoid outliers, mathematical programming formulations, complexity results},
author = {Nicolas Dupin and Frank Nielsen},
journal= {arXiv preprint arXiv:2302.05644},
year = {2023}
}