k-Median Clustering via Metric Embedding: Towards Better Initialization with Differential Privacy

Chenglin Fan, Ping Li, Xiaoyun Li

Keywords: k-Median Clustering, Differential Privacy

Abstract: In clustering algorithms, the choice of initial centers is crucial for the quality of the learned clusters. We propose a new initialization scheme for the $k$-median problem in the general metric space (e.g., discrete space induced by graphs), based on the construction of metric embedding tree structure of the data. From the tree, we propose a novel and efficient search algorithm, for good initial centers that can be used subsequently for the local search algorithm. The so-called HST initialization method can produce initial centers achieving lower errors than those from another popular initialization method, $k$-median++, with comparable efficiency. Our HST initialization can also be easily extended to the setting of differential privacy (DP) to generate private initial centers. We show that the error of applying DP local search followed by our private HST initialization improves previous results on the approximation error, and approaches the lower bound within a small factor. Experiments demonstrate the effectiveness of our proposed methods.

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