# Nearly-Tight and Oblivious Algorithms for Explainable Clustering

We study the problem of explainable clustering in the setting first formalized by Moshkovitz, Dasgupta, Rashtchian, and Frost (ICML 2020). A k-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the k leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of O(log^2 k) compared to an optimal (not necessarily explainable) clustering for the k-medians objective, and a factor of O(k log^2 k) for the k-means objective. This improves over the previous best upper bounds of O(k) and O(k^2), respectively, and nearly matches the previous Ω(log k) lower bound for k-medians and our new Ω(k) lower bound for k-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in ℝ^d, it is oblivious to the data points and runs in time O(dk log^2 k), independent of the number of data points n. Our upper and lower bounds also generalize to objectives given by higher ℓ_p-norms.

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