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

Abstract

In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (aij) with and a (small) k × k symmetric positive semidefinite matrix B = (bij). The goal is to find a partition {S1, …, Sk} of {1, … n} which maximizes . We design a polynomial time approximation algorithm that achieves an approximation ratio of , 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, …, vk ∊ ℝk then R(B) is the minimum radius of a Euclidean ball containing the points {v1, …, vk}. The parameter C(B) is defined as the maximum over all measurable partitions {A1, …, Ak} of ℝk–1 of the quantity , where for i ∊ {1, …, k} the vector zi ∊ ℝk–1 is the Gaussian moment of Ai, i.e., . We also show that for every ε > 0, achieving an approximation guarantee of is Unique Games hard.