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 2 d )-time algorithm by Eiben et al. [SODA’21] by a factor of n .