  K-means -- and the celebrated Lloyd algorithm -- is more than the clustering
method it was originally designed to be. It has indeed proven pivotal to help
increase the speed of many machine learning and data analysis techniques such
as indexing, nearest-neighbor search and prediction, data compression; its
beneficial use has been shown to carry over to the acceleration of kernel
machines (when using the Nystr\"om method). Here, we propose a fast extension
of K-means, dubbed QuicK-means, that rests on the idea of expressing the matrix
of the $K$ centroids as a product of sparse matrices, a feat made possible by
recent results devoted to find approximations of matrices as a product of
sparse factors. Using such a decomposition squashes the complexity of the
matrix-vector product between the factorized $K \times D$ centroid matrix
$\mathbf{U}$ and any vector from $\mathcal{O}(K D)$ to $\mathcal{O}(A \log
A+B)$, with $A=\min (K, D)$ and $B=\max (K, D)$, where $D$ is the dimension of
the training data. This drastic computational saving has a direct impact in the
assignment process of a point to a cluster, meaning that it is not only
tangible at prediction time, but also at training time, provided the
factorization procedure is performed during Lloyd's algorithm. We precisely
show that resorting to a factorization step at each iteration does not impair
the convergence of the optimization scheme and that, depending on the context,
it may entail a reduction of the training time. Finally, we provide discussions
and numerical simulations that show the versatility of our
computationally-efficient QuicK-means algorithm.
