Clustering lines in high-dimensional space: Classification of incomplete data

A set of κ balls B₁,. . ., B_κ in a Euclidean space is said to cover a collection of lines if every line intersects some ball.We consider the κ-center problem for lines in high-dimensional space: Given a set of n lines l = {l₁,. . ., l_n} in ℝ^d, find κ balls of minimum radius which cover l. We present a 2-approximation algorithm for the cases κ = 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space. Our result for 3-clustering is strongly based on a new result in discrete geometry that may be of independent interest: a Helly-type theorem for collections of axis-parallel "crosses" in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets. In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.