Simple, Scalable and Effective Clustering via One-Dimensional Projections

Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. Popular clustering algorithms such as Lloyd's algorithm and $k$-means++ can take $\Omega(ndk)$ time when clustering $n$ points in a $d$-dimensional space (represented by an $n\times d$ matrix $X$) into $k$ clusters. On massive datasets with moderate to large $k$, the multiplicative $k$ factor can become very expensive. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(\mathsf{nnz}(X) + n\log n)$ for arbitrary $k$. Here $\mathsf{nnz}(X)$ is the total number of non-zero entries in the input dataset $X$, which is upper bounded by $nd$ and can be significantly smaller for sparse datasets. We prove that our algorithm achieves approximation ratio $\widetilde{O}(k^4)$ on any input dataset for the $k$-means objective, and our experiments show that the quality of the clusters found by our algorithm is usually much better than this worst-case bound. We use our algorithm for $k$-means clustering and for coreset construction; our experiments show that it gives a new tradeoff between running time and cluster quality compared to previous state-of-the-art methods for these tasks. Our theoretical analysis is based on novel results of independent interest. We show that the approximation ratio achieved after a random one-dimensional projection can be lifted to the original points and that $k$-means++ seeding can be implemented in expected time $O(n\log n)$ in one dimension.