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

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 can extract good initial centers that can be used subsequently for the local search algorithm. Our method, named the HST initialization, can also be easily extended to the setting of differential privacy (DP) to generate private initial centers. Theoretically, the initial centers from HST initialization can achieve lower error than those from another popular initialization method, $k$-median++, in the non-DP setting. Moreover, with privacy constraint, we show that the error of applying DP local search followed by our private HST initialization improves previous results, and approaches the known lower bound within a small factor. Empirically, experiments are conducted to demonstrate the effectiveness of our methods.