Approximate kernel clustering

Subhash Khot, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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(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.

Original languageEnglish (US)
Title of host publicationProceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
Pages561-570
Number of pages10
DOIs
StatePublished - 2008
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: Oct 25 2008Oct 28 2008

Other

Other49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008
CountryUnited States
CityPhiladelphia, PA
Period10/25/0810/28/08

Fingerprint

Approximation algorithms
Learning systems
Computational complexity
Hardness
Polynomials

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Khot, S., & Naor, A. (2008). Approximate kernel clustering. In 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.

Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. p. 561-570 4690989.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Khot, S & Naor, A 2008, Approximate kernel clustering. in 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
Khot S, Naor A. Approximate kernel clustering. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. p. 561-570. 4690989 https://doi.org/10.1109/FOCS.2008.33
Khot, Subhash ; Naor, Assaf. / Approximate kernel clustering. Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008. 2008. pp. 561-570
@inproceedings{93b3faeb6d6f42f2aa9f8963caabd86f,
title = "Approximate kernel clustering",
abstract = "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.",
author = "Subhash Khot and Assaf Naor",
year = "2008",
doi = "10.1109/FOCS.2008.33",
language = "English (US)",
isbn = "9780769534367",
pages = "561--570",
booktitle = "Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008",

}

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 -