Approximate kernel clustering

Subhash Khot, Assaf Naor

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

8 Scopus citations

Abstract

In the kernel clustering problem we are given a large n × n positive semi-definite matrix A = (aij) with Σi,j=1 n 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 - Dec 30 2008
Externally publishedYes
Event49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008 - Philadelphia, PA, United States
Duration: Oct 25 2008Oct 28 2008

Publication series

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

Other

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

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

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  • 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] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2008.33