## Abstract

The k-means problem is to compute a set of k centers (points) that minimizes the sum of squared distances to a given set of n points in a metric space. Arguably, the most common algorithm to solve it is k-means++ which is easy to implement and provides a provably small approximation error in time that is linear in n. We generalize k-means++ to support outliers in two sense (simultaneously): (i) nonmetric spaces, e.g., M-estimators, where the distance dist(p, x) between a point p and a center x is replaced by min {dist(p, x), c} for an appropriate constant c that may depend on the scale of the input. (ii) k-means clustering with m ≥ 1 outliers, i.e., where the m farthest points from any given k centers are excluded from the total sum of distances. This is by using a simple reduction to the (k + m)-means clustering (with no outliers).

Original language | English |
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Article number | 311 |

Number of pages | 21 |

Journal | Algorithms |

Volume | 13 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2020 |

### Bibliographical note

Funding Information:Funding: Part of the research of this paper was sponsored by the Phenomix consortium of the Israel Innovation Authority. We thank them for this support.

Publisher Copyright:

© 2020 by the authors. Licensee MDPI, Basel, Switzerland.

## Keywords

- Approximation
- Clustering
- Outliers

## ASJC Scopus subject areas

- Theoretical Computer Science
- Numerical Analysis
- Computational Theory and Mathematics
- Computational Mathematics