TY - GEN

T1 - Approximate kernel clustering

AU - Khot, Subhash

AU - Naor, Assaf

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

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=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 (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=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 (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

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 561

EP - 570

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

T2 - 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008

Y2 - 25 October 2008 through 28 October 2008

ER -