Sharp kernel clustering algorithms and their associated Grothendieck inequalities

abstract Subhash Khot Assaf Naor In the kernel clustering problem we are given a (large) n ? n symmetric positive semidefinite matrix A = (aij) with n i=1 n j=1 aij = 0 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 k i=1 k j=1 (p,q)Si?Sj apq bij. 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, . . . , 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 k i=1 k j=1 bij zi, zj , where for i {1, . . . , k} the vector zi Rk-1 is the Gaussian moment of Ai, i.e., zi = 1 (2)(k-1)/2 Ai xe- x 2 2/2 dx. We al...

• Type: Conference
• Year: 2010
• Where: SODA
• Authors: Subhash Khot, Assaf Naor
• Keywords: Bij Zi, Discrete Algorithms, Positive Semidefinite Matrix, SODA 2010, Symmetric Positive Semidefinite