Near-optimal Algorithms for Explainable k-Medians and k-Means
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We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is Õ(log k) competitive with k-medians with ℓ_1 norm and Õ(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^3/2 k) competitive for k-medians with ℓ_2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω̃(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ_2 norm.
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