## Abstract

We study exact recovery conditions for convex relaxations of point cloud clustering problems, focusing on two of the most common optimization problems for unsupervised clustering: k-means and k-median clustering. Motivations for focusing on convex relaxations are: (a) they come with a certificate of optimality, and (b) they are generic tools which are relatively parameter-free, not tailored to specific assumptions over the input. More precisely, we consider the distributional setting where there are k clusters in R_{m} and data from each cluster consists of n points sampled from a symmetric distribution within a ball of unit radius. We ask: what is the minimal separation distance between cluster centers needed for convex relaxations to exactly recover these k clusters as the optimal integral solution? For the k-median linear programming relaxation we show a tight bound: exact recovery is obtained given arbitrarily small pairwise separation ε > 0 between the balls. In other words, the pairwise center separation is Δ > 2 +ε. Under the same distributional model, the k-means LP relaxation fails to recover such clusters at separation as large as Δ = 4. Yet, if we enforce PSD constraints on the k-means LP, we get exact cluster recovery at separation as low as Δ > min {2+ √2k/m, 2+ √2 + 2/m}+ε: In contrast, common heuristics such as Lloyd's algorithm (a.k.a. the k-means algorithm) can fail to recover clusters in this setting; even with arbitrarily large cluster separation, k-means++ with overseeding by any constant factor fails with high probability at exact cluster recovery. To complement the theoretical analysis, we provide an experimental study of the recovery guarantees for these various methods, and discuss several open problems which these experiments suggest.

Original language | English (US) |
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Title of host publication | ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science |

Publisher | Association for Computing Machinery, Inc |

Pages | 191-200 |

Number of pages | 10 |

ISBN (Electronic) | 9781450333337 |

DOIs | |

State | Published - Jan 11 2015 |

Event | 6th Conference on Innovations in Theoretical Computer Science, ITCS 2015 - Rehovot, Israel Duration: Jan 11 2015 → Jan 13 2015 |

### Publication series

Name | ITCS 2015 - Proceedings of the 6th Innovations in Theoretical Computer Science |
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### Other

Other | 6th Conference on Innovations in Theoretical Computer Science, ITCS 2015 |
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Country | Israel |

City | Rehovot |

Period | 1/11/15 → 1/13/15 |

## ASJC Scopus subject areas

- Computational Theory and Mathematics