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Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

EPTAS for k-means Clustering of Affine Subspaces

Abstract

We consider a generalization of the fundamental k-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in ℝd, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most Δ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most Δ, called a Δ-point. Thus we seek a partition of n input Δ-points into k clusters minimizing the k-means objective. For Δ = 0, when all coordinates of each point are specified, this is the usual k-means clustering. We give an algorithm that finds an (1 + )-approximate solution in time f(k, ∊, Δ) · n2 · d for some function f of k, ∊, and Δ only.

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cover image Proceedings
Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 2649 - 2659
Editor: Dániel Marx, CISPA Helmholtz Center for Information Security, Germany
ISBN (Online): 978-1-61197-646-5

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Published online: 7 January 2021

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A full version of this paper is available at https://arxiv.org/abs/2010.09580

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