### Abstract

In the kernel clustering problem we are given a large n × n positive semi-definite matrix A = (a_{ij}) with Σ_{i,j=1} ^{n} a_{ij} = 0 and a small k × k positive semi-definite matrix B = (b_{ij}). The goal is to find a partition S_{1},.. .,S_{k} of{1,...n} which maximizes the quantity Σ _{i,j=1}^{k} (Σ_{(i,j)∈Si×Sj} a _{ij}) b_{ij}. We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when B is the 3 × 3 identity matrix the UGC hardness threshold of this problem is exactly 16π/27. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when B is the k × k identity matrix is 8π/9 (1-1/k) for every k ≥ 3.

Original language | English (US) |
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Title of host publication | Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |

Pages | 561-570 |

Number of pages | 10 |

DOIs | |

State | Published - Dec 30 2008 |

Externally published | Yes |

Event | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States Duration: Oct 25 2008 → Oct 28 2008 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 |
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Country | United States |

City | Philadelphia, PA |

Period | 10/25/08 → 10/28/08 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)

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## Cite this

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008*(pp. 561-570). [4690989] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2008.33