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

TL;DR

This paper introduces a linear-time approximation scheme for k-means clustering of incomplete data represented as affine subspaces, significantly improving previous algorithms' efficiency.

Contribution

The paper presents a novel linear-time algorithm for k-means clustering of affine subspaces, reducing complexity from quadratic to linear in data size.

Findings

Achieves (1+ε)-approximate solutions in O(nd) time.

Constants depend only on Δ, ε, and k.

Improves previous O(n^2 d) algorithm by a factor of n.

Abstract

In this paper, we present a linear-time approximation scheme for $k$-means clustering of \emph{incomplete} data points in $d$-dimensional Euclidean space. An \emph{incomplete} data point with $\Delta>0$ unspecified entries is represented as an axis-parallel affine subspaces of dimension $\Delta$. 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 $\Delta$ that yields an $(1+\epsilon)$-approximate solution in $O(nd)$ time. The constants hidden behind $O(\cdot)$ depend only on $\Delta, \epsilon$ and $k$. This improves the $O(n^2 d)$-time algorithm by Eiben et al.[SODA'21] by a factor of $n$.