  Kernel $k$-means clustering can correctly identify and extract a far more
varied collection of cluster structures than the linear $k$-means clustering
algorithm. However, kernel $k$-means clustering is computationally expensive
when the non-linear feature map is high-dimensional and there are many input
points. Kernel approximation, e.g., the Nystr\"om method, has been applied in
previous works to approximately solve kernel learning problems when both of the
above conditions are present. This work analyzes the application of this
paradigm to kernel $k$-means clustering, and shows that applying the linear
$k$-means clustering algorithm to $\frac{k}{\epsilon} (1 + o(1))$ features
constructed using a so-called rank-restricted Nystr\"om approximation results
in cluster assignments that satisfy a $1 + \epsilon$ approximation ratio in
terms of the kernel $k$-means cost function, relative to the guarantee provided
by the same algorithm without the use of the Nystr\"om method. As part of the
analysis, this work establishes a novel $1 + \epsilon$ relative-error trace
norm guarantee for low-rank approximation using the rank-restricted Nystr\"om
approximation. Empirical evaluations on the $8.1$ million instance MNIST8M
dataset demonstrate the scalability and usefulness of kernel $k$-means
clustering with Nystr\"om approximation. This work argues that spectral
clustering using Nystr\"om approximation---a popular and computationally
efficient, but theoretically unsound approach to non-linear clustering---should
be replaced with the efficient and theoretically sound combination of kernel
$k$-means clustering with Nystr\"om approximation. The superior performance of
the latter approach is empirically verified.
