### Abstract

In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (a_{ij}) with ∑^{n}
_{i=1} ∑^{n}
_{j=1} a_{ij}=0 and a (small) k × k symmetric positive semidefinite matrix B = (b_{ij}). The goal is to find a partition {S_{1},...,S_{k}} of {1,...n} which maximizes ∑^{k}
_{i=1} ∑^{k}
_{i=1} (∑_{(p,q)εSi×Sj} a_{pq})b_{ij}. We design a polynomial time approximation algorithm that achieves an approximation ratio of R(B)^{2}/C(B), where R(B) and C(B) are geometric parameters that depend only on the matrix B, defined as follows: if b_{ij} = 〈v_{i},v_{j}〉 is the Gram matrix representation of B for some v_{1},....,v_{k}εℝ^{k} then R(B) is the minimum radius of a Euclidean ball containing the points {v_{1},...,v_{k}}. The parameter C(B) is defined as the maximum over all measurable partitions {A_{1},...,A_{k}} of ℝ^{k-1} of the quantity ∑^{k}
_{i=1} ∑^{k}
_{j=1}n_{ij} 〈Z_{i}, Z_{j}〉where for iε{1,...,k} the vector Z_{i} ε ℝ^{k-1} is the Gaussian moment of A_{i}. We also show that for every ε > 0, achieving an approximation guarantee of is Unique Games hard.

Original language | English (US) |
---|---|

Pages (from-to) | 269-300 |

Number of pages | 32 |

Journal | Random Structures and Algorithms |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - May 2013 |

### Fingerprint

### Keywords

- Approximation algorithms
- Semidefinite programming
- Unique Games hardness

### ASJC Scopus subject areas

- Computer Graphics and Computer-Aided Design
- Software
- Mathematics(all)
- Applied Mathematics

### Cite this

*Random Structures and Algorithms*,

*42*(3), 269-300. https://doi.org/10.1002/rsa.20398

**Sharp kernel clustering algorithms and their associated Grothendieck inequalities.** / Khot, Subhash; Naor, Assaf.

Research output: Contribution to journal › Article

*Random Structures and Algorithms*, vol. 42, no. 3, pp. 269-300. https://doi.org/10.1002/rsa.20398

}

TY - JOUR

T1 - Sharp kernel clustering algorithms and their associated Grothendieck inequalities

AU - Khot, Subhash

AU - Naor, Assaf

PY - 2013/5

Y1 - 2013/5

N2 - In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (aij) with ∑n i=1 ∑n j=1 aij=0 and a (small) k × k symmetric positive semidefinite matrix B = (bij). The goal is to find a partition {S1,...,Sk} of {1,...n} which maximizes ∑k i=1 ∑k i=1 (∑(p,q)εSi×Sj apq)bij. We design a polynomial time approximation algorithm that achieves an approximation ratio of R(B)2/C(B), where R(B) and C(B) are geometric parameters that depend only on the matrix B, defined as follows: if bij = 〈vi,vj〉 is the Gram matrix representation of B for some v1,....,vkεℝk then R(B) is the minimum radius of a Euclidean ball containing the points {v1,...,vk}. The parameter C(B) is defined as the maximum over all measurable partitions {A1,...,Ak} of ℝk-1 of the quantity ∑k i=1 ∑k j=1nij 〈Zi, Zj〉where for iε{1,...,k} the vector Zi ε ℝk-1 is the Gaussian moment of Ai. We also show that for every ε > 0, achieving an approximation guarantee of is Unique Games hard.

AB - In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (aij) with ∑n i=1 ∑n j=1 aij=0 and a (small) k × k symmetric positive semidefinite matrix B = (bij). The goal is to find a partition {S1,...,Sk} of {1,...n} which maximizes ∑k i=1 ∑k i=1 (∑(p,q)εSi×Sj apq)bij. We design a polynomial time approximation algorithm that achieves an approximation ratio of R(B)2/C(B), where R(B) and C(B) are geometric parameters that depend only on the matrix B, defined as follows: if bij = 〈vi,vj〉 is the Gram matrix representation of B for some v1,....,vkεℝk then R(B) is the minimum radius of a Euclidean ball containing the points {v1,...,vk}. The parameter C(B) is defined as the maximum over all measurable partitions {A1,...,Ak} of ℝk-1 of the quantity ∑k i=1 ∑k j=1nij 〈Zi, Zj〉where for iε{1,...,k} the vector Zi ε ℝk-1 is the Gaussian moment of Ai. We also show that for every ε > 0, achieving an approximation guarantee of is Unique Games hard.

KW - Approximation algorithms

KW - Semidefinite programming

KW - Unique Games hardness

UR - http://www.scopus.com/inward/record.url?scp=84875547836&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875547836&partnerID=8YFLogxK

U2 - 10.1002/rsa.20398

DO - 10.1002/rsa.20398

M3 - Article

VL - 42

SP - 269

EP - 300

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -