Simple, Scalable and Effective Clustering via One-Dimensional Projections

• URL: http://arxiv.org/abs/2310.16752v1
• Date: Wed, 25 Oct 2023 16:37:45 GMT
• Title: Simple, Scalable and Effective Clustering via One-Dimensional Projections
• Authors: Moses Charikar, Monika Henzinger, Lunjia Hu, Maxmilian V\"otsch, Erik Waingarten
• Abstract summary: Clustering is a fundamental problem in unsupervised machine learning with many applications in data analysis. We introduce a simple randomized clustering algorithm that provably runs in expected time $O(mathrmnnz(X) + nlog n)$ for arbitrary $k$. We prove that our algorithm achieves approximation ratio $smashwidetildeO(k4)$ on any input dataset for the $k$-means objective.
• Score: 10.807367640692021
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: 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. In applications 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 $O(\mathrm{nnz}(X) + n\log n)$ for arbitrary $k$. Here $\mathrm{nnz}(X)$ 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 $\smash{\widetilde{O}(k^4)}$ on any input dataset for the $k$-means objective. We also believe that our theoretical analysis is of independent interest, as we show that the approximation ratio of a $k$-means algorithm is approximately preserved under a class of projections and that $k$-means++ seeding can be implemented in expected $O(n \log n)$ time in one dimension. Finally, we show experimentally that our clustering algorithm gives a new tradeoff between running time and cluster quality compared to previous state-of-the-art methods for these tasks.

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