Random Projections for $k$-means Clustering

This paper discusses the topic of dimensionality reduction for k-means clustering. We prove that any set of n points in d dimensions (rows in a matrix A ∈ R) can be projected into t = Ω(k/ε) dimensions, for any ε ∈ (0, 1/3), in O(nd⌈ε−2k/ log(d)⌉) time, such that with constant probability the optimal k-partition of the point set is preserved within a factor of 2 + ε. The projection is done by post-multiplying A with a d × t random matrix R having entries +1/ √ t or −1/ √ t with equal probability. A numerical implementation of our technique and experiments on a large face images dataset verify the speed and the accuracy of our theoretical results.