Linear-Time Approximation Scheme for k-Means Clustering of Affine Subspaces
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In this paper, we present a linear-time approximation scheme for k-means clustering of incomplete data points in d-dimensional Euclidean space. An incomplete data point with Δ>0 unspecified entries is represented as an axis-parallel affine subspaces of dimension Δ. The distance between two incomplete data points is defined as the Euclidean distance between two closest points in the axis-parallel affine subspaces corresponding to the data points. We present an algorithm for k-means clustering of axis-parallel affine subspaces of dimension Δ that yields an (1+ϵ)-approximate solution in O(nd) time. The constants hidden behind O(·) depend only on Δ, ϵ and k. This improves the O(n^2 d)-time algorithm by Eiben et al.[SODA'21] by a factor of n.
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