### Abstract

Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1 + ε) under a projection onto a random O(log(k/ε)/ε^{2})-dimensional subspace. Further, the cost of every clustering is preserved within (1 + ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p. For k-means, our result resolves an open problem posed by Cohen, Elder, Musco, Musco, and Persu (STOC 2015); for k-medians, it answers a question raised by Kannan.

Original language | English (US) |
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Title of host publication | STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing |

Editors | Moses Charikar, Edith Cohen |

Publisher | Association for Computing Machinery |

Pages | 1027-1038 |

Number of pages | 12 |

ISBN (Electronic) | 9781450367059 |

DOIs | |

State | Published - Jun 23 2019 |

Event | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 - Phoenix, United States Duration: Jun 23 2019 → Jun 26 2019 |

### Publication series

Name | Proceedings of the Annual ACM Symposium on Theory of Computing |
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ISSN (Print) | 0737-8017 |

### Conference

Conference | 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 |
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Country | United States |

City | Phoenix |

Period | 6/23/19 → 6/26/19 |

### Keywords

- Clustering
- Dimension reduction
- Johnson-Lindenstrauss transform
- K-means
- K-medians
- Kirszbraun theorem

### ASJC Scopus subject areas

- Software

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## Cite this

*STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing*(pp. 1027-1038). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3313276.3316350