A Single-Swap Local Search Algorithm for k-Means of Lines

Clustering is a fundamental problem that has been extensively studied over past few decades, with most research focusing on point-based clustering such as $k$-means, $k$-median, and $k$-center. However, numerous real-world applications, such as motion analysis, computer vision, and missing data analysis, require clustering over structured data, including lines, time series and affine subspaces (flats), where traditional point-based clustering algorithms often fall short. In this paper, we study the $k$-means of lines problem, where the input is a set $L$ of lines in $\mathbb{R}^d$, and the goal is to find $k$ centers $C$ in $\mathbb{R}^d$ such that the sum of squared distances from each line in $L$ to its nearest center in $C$ is minimized. The local search algorithm is a well-established strategy for point-based $k$-means clustering, known for its efficiency and provable approximation guarantees. However, extending local search algorithm to the $k$-means of lines problem is nontrivial, as the capture relation used in point-based clustering does not generalize to the line setting.