Random Projections for k-means Clustering

We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \in \RR {n \times d}$) can be projected into $t \Omega(k / \eps 2)$ dimensions, for any $\eps \in (0,1/3)$, in $O(n d \lceil \eps {-2} k/ \log(d) \rceil)$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2 \eps$. The projection is done by post-multiplying $A$ with a $d \times t$ random matrix $R$ having entries $ 1/\sqrt{t}$ or $-1/\sqrt{t}$ 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. Papers published at the Neural Information Processing Systems Conference.