Sharp kernel clustering algorithms and their associated Grothendieck inequalities: Extended abstract [chapter]

Subhash Khot, Assaf Naor
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (a ij ) with 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, . . . , v k ∈ R k then R(B) is the minimum radius of a Euclidean ball containing the points {v 1 , . . . , v
more » ... k }. The parameter C(B) is defined as the maximum over all . . , k} the vector zi ∈ R k−1 is the Gaussian moment of Ai, i.e., zi = 1 (2π) (k−1)/2 A i xe − x 2 2 /2 dx. We also show that for every ε > 0, achieving an approximation guarantee of (1 − ε) R(B) 2 C(B) is Unique Games hard.
doi:10.1137/1.9781611973075.55 dblp:conf/soda/KhotN10 fatcat:jr7din5bknaqfe46mownnliv4e