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Approximate Kernel Clustering

Part of: Algorithms - Computer Science

Published online by Cambridge University Press:  21 December 2009

Subhash Khot  and
Assaf Naor
Subhash Khot
Affiliation:
Subhash Khot, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A., E-mail: [email protected]
Assaf Naor
Affiliation:
Assaf Naor, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A., E-mail: [email protected]
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Abstract

In the kernel clustering problem we are given a large n × n positive-semidefinite matrix A = (aij) with and a small k × k positive-semidefinite matrix B = (bij). The goal is to find a partition S1, …, Sk of {1, … n} which maximizes the quantity

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 et al. 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.

MSC classification

Secondary: 68W25: Approximation algorithms
Type
Research Article
Information
Mathematika , Volume 55 , Issue 1-2 , December 2009 , pp. 129 - 165
Copyright
Copyright © University College London 2009

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