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

TL;DR

This paper introduces a Partial k-means variant to handle outliers, providing mathematical programming formulations, complexity analysis, and efficient solutions for one-dimensional cases.

Contribution

It extends MSSC by considering outliers, offers integer programming formulations, and analyzes complexity, including polynomial solutions for 1D cases.

Findings

PMSSC is NP-hard in Euclidean space for dimensions > 2

Unweighted PMSSC is polynomial in 1D and solved with dynamic programming

Weighted PMSSC has a weaker optimality property, complexity remains open

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.