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

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 subspace 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(n² d)-time algorithm by Eiben et al. [SODA'21] by a factor of n.