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• W3175038243 abstract "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 $Omega(log k)$ lower bound for $k$-medians and our new $Omega(k)$ lower bound for $k$-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in $mathbb{R}^d$, 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 $ell_p$-norms." @default.
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• W3175038243 date "2021-06-30" @default.
• W3175038243 modified "2023-10-01" @default.
• W3175038243 title "Nearly-Tight and Oblivious Algorithms for Explainable Clustering" @default.
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