Global optimality in k-means clustering

Abstract: We study the problem of finding an optimum clustering, a problem known to be NP-hard. Existing literature contains algorithms running in time proportional to the number of points raised to a power that depends on the dimensionality and on the number of clusters. Published validations of some of these algorithms are unfortunately incomplete; besides, the constant factors (with respect to the number of points) in their running time bounds have seen several published important improvements but are still huge, exponential on the dimension and on the number of clusters, making the corresponding algorithms fully impractical. We provide a new algorithm, with its corresponding complexity-theoretic analysis. It reduces both the exponent and the constant factor, to the extent that it becomes feasible for relevant particular cases. Additionally, it parallelizes extremely well, so that its implementation on current high-performance hardware is quite straightforward. Our proposal opens the door to potential improvements along a research line that had no practical significance so far; besides, a long but single-shot run of our algorithm allows one to identify absolutely optimum solutions for benchmark problems, whereby alternative heuristic proposals can evaluate the goodness of their solutions and the precise price paid for their faster running times

U-Control Chart Based Differential Evolution Clustering for Determining the Number of Cluster in k-Means

The automatic clustering differential evolution (ACDE) is one of the clustering methods that are able to determine the cluster number automatically. However, ACDE still makes use of the manual strategy to determine k activation threshold thereby affecting its performance. In this study, the ACDE problem will be ameliorated using the u-control chart (UCC) then the cluster number generated from ACDE will be fed to k-means. The performance of the proposed method was tested using six public datasets from the UCI repository about academic efficiency (AE) and evaluated with Davies Bouldin Index (DBI) and Cosine Similarity (CS) measure. The results show that the proposed method yields excellent performance compared to prior researches

Global optimality in k-means clustering

We study the problem of finding an optimum clustering, a problem known to be NP-hard. Existing literature contains algorithms running in time proportional to the number of points raised to a power that depends on the dimensionality and on the number of clusters. Published validations of some of these algorithms are unfortunately incomplete; besides, the constant factors (with respect to the number of points) in their running time bounds have seen several published important improvements but are still huge, exponential on the dimension and on the number of clusters, making the corresponding algorithms fully impractical. We provide a new algorithm, with its corresponding complexity-theoretic analysis. It reduces both the exponent and the constant factor, to the extent that it becomes feasible for relevant particular cases. Additionally, it parallelizes extremely well, so that its implementation on current high-performance hardware is quite straightforward. Our proposal opens the door to potential improvements along a research line that had no practical significance so far; besides, a long but single-shot run of our algorithm allows one to identify absolutely optimum solutions for benchmark problems, whereby alternative heuristic proposals can evaluate the goodness of their solutions and the precise price paid for their faster running times.Peer Reviewe