Linear-Time Approximation Scheme for k-Means Clustering of Axis-Parallel Affine Subspaces

Abstract

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 ∆ > 0 unspecified entries is represented as an axis-parallel affine subspace of dimension ∆. 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 ∆ that yields an (1 + ϵ)-approximate solution in O(nd) time. The constants hidden behind O(·) depend only on ∆, ϵ and k. This improves the O(n