Nearly-Tight and Oblivious Algorithms for Explainable Clustering

• URL: http://arxiv.org/abs/2106.16147v1
• Date: Wed, 30 Jun 2021 15:49:41 GMT
• Title: Nearly-Tight and Oblivious Algorithms for Explainable Clustering
• Authors: Buddhima Gamlath, Xinrui Jia, Adam Polak, Ola Svensson
• Abstract summary: 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 data points with a threshold cut in a single dimension (feature) We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(log2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective.
• Score: 8.071379672971542
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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