Sharp kernel clustering algorithms and their associated Grothendieck inequalities

In the kernel clustering problem we are given a (large) n × n symmetric positive semidefinite matrix A = (a_ij) with \begin{align*}\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0\end{align*} and a (small) k × k symmetric positive semidefinite matrix B = (b_ij). The goal is to find a partition {S₁,…,S_k} of {1,…n} which maximizes \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*}. We design a polynomial time approximation algorithm that achieves an approximation ratio of \begin{align*}\frac{R(B)^2}{C(B)}\end{align*}, where R(B) and C(B) are geometric parameters that depend only on the matrix B, defined as follows: if b_ij = 〈v_i,v_j〉 is the Gram matrix representation of B for some \begin{align*}v_1,\ldots,v_k\in \mathbb{R}^k\end{align*} then R(B) is the minimum radius of a Euclidean ball containing the points {v₁,…,v_k}. The parameter C(B) is defined as the maximum over all measurable partitions {A₁,…,A_k} of \begin{align*}\mathbb{R}^{k-1}\end{align*} of the quantity \begin{align*}\sum_{i=1}^k\sum_{j=1}^k b_{ij}\langle z_i,z_j\rangle\end{align*}, where for i∈{1,…,k} the vector \begin{align*}z_i\in \mathbb{R}^{k-1}\end{align*} is the Gaussian moment of A_i, i.e., \begin{align*}z_i=\frac{1}{(2\pi)^{(k-1)/2}}\int_{A_i}xe^{-\|x\|_2^2/2}dx\end{align*}. We also show that for every ε > 0, achieving an approximation guarantee of \begin{align*}(1-\varepsilon)\frac{R(B)^2}{C(B)}\end{align*} is Unique Games hard. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013