  Spectral clustering is one of the most important algorithms in data mining
and machine intelligence; however, its computational complexity limits its
application to truly large scale data analysis. The computational bottleneck in
spectral clustering is computing a few of the top eigenvectors of the
(normalized) Laplacian matrix corresponding to the graph representing the data
to be clustered. One way to speed up the computation of these eigenvectors is
to use the "power method" from the numerical linear algebra literature.
Although the power method has been empirically used to speed up spectral
clustering, the theory behind this approach, to the best of our knowledge,
remains unexplored. This paper provides the \emph{first} such rigorous
theoretical justification, arguing that a small number of power iterations
suffices to obtain near-optimal partitionings using the approximate
eigenvectors. Specifically, we prove that solving the $k$-means clustering
problem on the approximate eigenvectors obtained via the power method gives an
additive-error approximation to solving the $k$-means problem on the optimal
eigenvectors.
