K-means+++: Outliers-resistant clustering

Adiel Statman, Liat Rozenberg, Dan Feldman

Research output: Contribution to journalArticlepeer-review


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 languageEnglish
Article number311
Number of pages21
Issue number12
StatePublished - 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.


  • Approximation
  • Clustering
  • Outliers

ASJC Scopus subject areas

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


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