k^2 k 2 -means for Fast and Accurate Large Scale Clustering

We propose \(k^2\)-means, a new clustering method which efficiently copes with large numbers of clusters and achieves low energy solutions. \(k^2\)-means builds upon the standard k-means (Lloyd’s algorithm) and combines a new strategy to accelerate the convergence with a new low time complexity divisive initialization. The accelerated convergence is achieved through only looking at \(k_n\) nearest clusters and using triangle inequality bounds in the assignment step while the divisive initialization employs an optimal 2-clustering along a direction. The worst-case time complexity per iteration of our \(k^2\)-means is \(O(nk_nd\,+\,k^2d)\), where d is the dimension of the n data points and k is the number of clusters and usually \(n\gg k \gg k_n\). Compared to k-means’ O(nkd) complexity, our \(k^2\)-means complexity is significantly lower, at the expense of slightly increasing the memory complexity by \(O(nk_n+k^2)\). In our extensive experiments \(k^2\)-means is order(s) of magnitude faster than standard methods in computing accurate clusterings on several standard datasets and settings with hundreds of clusters and high dimensional data. Moreover, the proposed divisive initialization generally leads to clustering energies comparable to those achieved with the standard k-means++ initialization, while being significantly faster.