Coresets for Clustering with Missing Values
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We provide the first coreset for clustering points in ℝ^d that have multiple missing values (coordinates). Previous coreset constructions only allow one missing coordinate. The challenge in this setting is that objective functions, like k-Means, are evaluated only on the set of available (non-missing) coordinates, which varies across points. Recall that an ϵ-coreset of a large dataset is a small proxy, usually a reweighted subset of points, that (1+ϵ)-approximates the clustering objective for every possible center set. Our coresets for k-Means and k-Median clustering have size (jk)^O(min(j,k)) (ϵ^-1 d log n)^2, where n is the number of data points, d is the dimension and j is the maximum number of missing coordinates for each data point. We further design an algorithm to construct these coresets in near-linear time, and consequently improve a recent quadratic-time PTAS for k-Means with missing values [Eiben et al., SODA 2021] to near-linear time. We validate our coreset construction, which is based on importance sampling and is easy to implement, on various real data sets. Our coreset exhibits a flexible tradeoff between coreset size and accuracy, and generally outperforms the uniform-sampling baseline. Furthermore, it significantly speeds up a Lloyd's-style heuristic for k-Means with missing values.
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