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 $\Delta >0$ 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 n axis-parallel affine subspaces of dimension Δ that yields an $(1+ϵ)$-approximate solution in $O\left(nd\right)$ time. The constants hidden behind $O(\cdot )$ depend only on $\Delta ,ϵ$ and k. This improves the $O\left(n^{2}d\right)$-time algorithm by Eiben et al. (2021) [7] by a factor of n.