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# Near-Linear Time Approximation Algorithms for k-means with Outliers

*Proceedings of the 41st International Conference on Machine Learning*, PMLR 235:19723-19756, 2024.

#### Abstract

The k-means with outliers problem is one of the most extensively studied clustering problems in the field of machine learning, where the goal is to discard up to z outliers and identify a minimum k-means clustering on the remaining data points. Most previous results for this problem have running time dependent on the aspect ratio Δ (the ratio between the maximum and the minimum pairwise distances) to achieve fast approximations. To address the issue of aspect ratio dependency on the running time, we propose sampling-based algorithms with almost linear running time in the data size, where a crucial component of our approach is an algorithm called Fast-Sampling. Fast-Sampling algorithm can find inliers that well approximate the optimal clustering centers without relying on a guess for the optimal clustering costs, where a 4-approximate solution can be obtained in time $O(\frac{ndk\log\log n}{\epsilon^2})$ with O(k/ϵ) centers opened and (1+ϵ)z outliers discarded. To reduce the number of centers opened, we propose a center reduction algorithm, where an O(1/ϵ)-approximate solution can be obtained in time $O(\frac{ndk\log \log n}{\epsilon^2} + dpoly(k, \frac{1}{\epsilon})\log(n\Delta))$ with (1+ϵ)z outliers discarded and exactly k centers opened. Empirical experiments suggest that our proposed sampling-based algorithms outperform state-of-the-art algorithms for the k-means with outliers problem.