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 $\mathrm{\Delta }>0$ 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 $O(nd)$ time. The constants hidden behind $O(\cdot )$ depend only on $\mathrm{\Delta },ϵ$ and k. This improves the $O(n^{2}d)$-time algorithm by Eiben et al. (2021) [7] by a factor of n.

在本文中，我们针对d维欧几里得空间中的不完整数据点的k均值聚类提出了线性时间近似方案。一个不完整的数据点$\mathrm{△}>0$未指定的条目表示为维度为 Δ 的轴平行仿射子空间。两个不完整数据点之间的距离定义为与数据点对应的轴平行仿射子空间中两个最近点之间的欧氏距离。我们提出了一种算法，用于对维度为 Δ 的n轴平行仿射子空间进行k均值聚类，从而产生$(\mathrm{1个}+\epsilon )$-近似解$欧(nd)$时间。背后隐藏的常量$欧(\cdot )$只依赖于$\mathrm{△},\epsilon$和k。这提高了$欧(n^{\mathrm{2个}}d)$-Eiben 等人的时间算法。(2021) [7]乘以n倍。