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Published November 2010 | public
Journal Article

Clustering Lines in High-Dimensional Space: Classification of Incomplete Data


A set of k balls B_1,...,B_k in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the k-center problem for lines in high-dimensional space: Given a set of n lines ^I= {I_1,...,l_n in R^d, find k balls of minimum radius which cover I. We present a 2-approximation algorithm for the cases k = 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.

Additional Information

© 2010 ACM. Received Janauary 2008; Revised June 2009; Accepted December 2009. Publication date: November 2010. Part of this work was done when J. Gaowas a postdoctoral scholar with the Center for the Mathematics of Information, California Institute of Technology. Part of this work was done when M. Langberg was a postdoctoral scholar at the California Institute of Technology. The research of M. Langberg was supported in part by NSF grant CCF-0346991. The research of L. J. Schulman was supported in part by NSF CCF-0515342, NSA H98230-06-1-0074, and NSF ITR CCR-0326554. Michael Langberg would like to thank Dan Feldman for sharing ideas that led to the proof presented in Section B.1.

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