Approximate Kernel Clustering

In the kernel clustering problem we are given a large ntimesn positive semi-definite matrix A=(a_ij) with Sigma_i,j ⁿ=1 a_ij=0 and a small ktimesk positivesemi-definite matrix B=b_ij. The goal is to find a partition S₁,..S_k of {1,...n} which maximizes the quantity Sigma_i,j=1 ^k(Sigma_{(i,j)isinS i timesS j}). We study the computational complexity of this generic clustering problem which originates in the theory of machine learning. We design a constant factor polynomial time approximation algorithm forthis problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manage to compute the sharp approximation threshold for this problem assuming the unique games conjecture (UGC). In particular, when B is the 3times3 identity matrix the UGC hardness threshold of this problem is exactly 16pi/27. We present and study a geometricconjecture of independent interest which we show would imply thatthe UGC threshold when B is the ktimesk identity matrix is 8pi/9(1-1/k) for every kges3.