Sharp kernel clustering algorithms and their associated Grothendieck inequalities

Abstract

AbstractIn the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (aij) with \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0\end{align*}\end{document} 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 \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\sum_{i=1}^k\sum_{j=1}^k \left(\sum_{(p,q)\in S_i\times S_j}a_{pq}\right)b_{ij}\end{align*}\end{document}. We design a polynomial time approximation algorithm that achieves an approximation ratio of \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\frac{R(B)^2}{C(B)}\end{align*}\end{document}, 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 \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}v_1,\ldots,v_k\in \mathbb{R}^k\end{align*}\end{document} 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 \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\mathbb{R}^{k-1}\end{align*}\end{document} of the quantity \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}\sum_{i=1}^k\sum_{j=1}^k b_{ij}\langle z_i,z_j\rangle\end{align*}\end{document}, where for i∈{1,…,k} the vector \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}z_i\in \mathbb{R}^{k-1}\end{align*}\end{document} is the Gaussian moment of Ai, i.e., \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}z_i=\frac{1}{(2\pi)^{(k-1)/2}}\int_{A_i}xe^{-\|x\|_2^2/2}dx\end{align*}\end{document}. We also show that for every ε > 0, achieving an approximation guarantee of \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}(1-\varepsilon)\frac{R(B)^2}{C(B)}\end{align*}\end{document} is Unique Games hard. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013