This paper considers $k$-means clustering in the presence of noise. It is known that $k$-means clustering is highly sensitive to noise, and thus noise should be removed to obtain a quality solution. A popular formulation of this problem is called $k$-means clustering with outliers. The goal of $k$-means clustering with outliers is to discard up to a specified number $z$ of points as noise/outliers and then find a $k$-means solution on the remaining data. The problem has received significant attention, yet current algorithms with theoretical guarantees suffer from either high running time or inherent loss in the solution quality. The main contribution of this paper is two-fold. Firstly, we develop a simple greedy algorithm that has provably strong worst case guarantees. The greedy algorithm adds a simple preprocessing step to remove noise, which can be combined with any $k$-means clustering algorithm. This algorithm gives the first pseudo-approximation-preserving reduction from $k$-means with outliers to $k$-means without outliers. Secondly, we show how to construct a coreset of size $O(k log n)$. When combined with our greedy algorithm, we obtain a scalable, near linear time algorithm. The theoretical contributions are verified experimentally by demonstrating that the algorithm quickly removes noise and obtains a high-quality clustering.
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