### 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 - 2008 |

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

### 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 |

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### ASJC Scopus subject areas

- Computer Science(all)

### Cite this

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

**Approximate kernel clustering.** / Khot, Subhash; Naor, Assaf.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008.*, 4690989, pp. 561-570, 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, Philadelphia, PA, United States, 10/25/08. https://doi.org/10.1109/FOCS.2008.33

}

TY - GEN

T1 - Approximate kernel clustering

AU - Khot, Subhash

AU - Naor, Assaf

PY - 2008

Y1 - 2008

N2 - In the kernel clustering problem we are given a large n × n positive semi-definite matrix A = (aij) with Σi,j=1n aij = 0 and a small k × k positive semi-definite matrix B = (bij). The goal is to find a partition S1,.. .,Sk of{1,...n} which maximizes the quantity Σ i,j=1k (Σ(i,j)∈Si×Sj a ij) bij. 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.

AB - In the kernel clustering problem we are given a large n × n positive semi-definite matrix A = (aij) with Σi,j=1n aij = 0 and a small k × k positive semi-definite matrix B = (bij). The goal is to find a partition S1,.. .,Sk of{1,...n} which maximizes the quantity Σ i,j=1k (Σ(i,j)∈Si×Sj a ij) bij. 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.

UR - http://www.scopus.com/inward/record.url?scp=57949086638&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57949086638&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2008.33

DO - 10.1109/FOCS.2008.33

M3 - Conference contribution

AN - SCOPUS:57949086638

SN - 9780769534367

SP - 561

EP - 570

BT - Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

ER -