  This paper studies the optimality of kernel methods in high-dimensional data
clustering. Recent works have studied the large sample performance of kernel
clustering in the high-dimensional regime, where Euclidean distance becomes
less informative. However, it is unknown whether popular methods, such as
kernel k-means, are optimal in this regime. We consider the problem of
high-dimensional Gaussian clustering and show that, with the exponential kernel
function, the sufficient conditions for partial recovery of clusters using the
NP-hard kernel k-means objective matches the known information-theoretic limit
up to a factor of $\sqrt{2}$ for large $k$. It also exactly matches the known
upper bounds for the non-kernel setting. We also show that a semi-definite
relaxation of the kernel k-means procedure matches up to constant factors, the
spectral threshold, below which no polynomial-time algorithm is known to
succeed. This is the first work that provides such optimality guarantees for
the kernel k-means as well as its convex relaxation. Our proofs demonstrate the
utility of the less known polynomial concentration results for random variables
with exponentially decaying tails in a higher-order analysis of kernel methods.
