  We study the topic of dimensionality reduction for $k$-means clustering.
Dimensionality reduction encompasses the union of two approaches: \emph{feature
selection} and \emph{feature extraction}. A feature selection based algorithm
for $k$-means clustering selects a small subset of the input features and then
applies $k$-means clustering on the selected features. A feature extraction
based algorithm for $k$-means clustering constructs a small set of new
artificial features and then applies $k$-means clustering on the constructed
features. Despite the significance of $k$-means clustering as well as the
wealth of heuristic methods addressing it, provably accurate feature selection
methods for $k$-means clustering are not known. On the other hand, two provably
accurate feature extraction methods for $k$-means clustering are known in the
literature; one is based on random projections and the other is based on the
singular value decomposition (SVD).
  This paper makes further progress towards a better understanding of
dimensionality reduction for $k$-means clustering. Namely, we present the first
provably accurate feature selection method for $k$-means clustering and, in
addition, we present two feature extraction methods. The first feature
extraction method is based on random projections and it improves upon the
existing results in terms of time complexity and number of features needed to
be extracted. The second feature extraction method is based on fast approximate
SVD factorizations and it also improves upon the existing results in terms of
time complexity. The proposed algorithms are randomized and provide
constant-factor approximation guarantees with respect to the optimal $k$-means
objective value.
