Local $K$-means: An Efficient Optimization Algorithm And Its Generalization
Until now, $k$-means is still one of the most popular clustering algorithms because of its simplicity and efficiency, although it has been proposed for a long time. In this paper, we considered a variant of $k$-means that takes the $k$-nearest neighbor ($k$-NN) graph as input and proposed a novel clustering algorithm called Local K-Means (LKM). We also developed a general model that unified LKM, KSUMS, and SC, and discussed the connection among them. In addition, we proposed an efficient optimization algorithm for the unified model. Thus, not only LKM but also SC can be optimized with a linear time complexity with respect to the number of samples. Specifically, the computational overhead is $O(nk)$, where $n$ and $k$ are denote the number of samples and nearest neighbors, respectively. Extensive experiments have been conducted on 11 synthetic and 16 benchmark datasets from the literature. The effectiveness, efficiency, and robustness to outliers of the proposed method have been verified by the experimental results.
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