## Abstract

The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in ℝ^{d} , incomplete data objects correspond to affine subspaces (lines or Δ-flats). With this motivation we study the problem of finding the minimum intersection radius r(L) of a set of lines or Δ-flats L: the least r such that there is a ball of radius r intersecting every flat in L. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher-dimensional flats, primarily because "distances" between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly's theorem. This "intrinsic-dimension" Helly theorem states: for any family L of Δ-dimensional convex sets in a Hilbert space, there exist Δ+2 sets L′ ⊆ L such that r(L)≤2r(L′). Based upon this we present an algorithm that computes a (1+ε)-core set L′ ⊆ L, |L′|=O(Δ^{4}/ε), such that the ball centered at a point c with radius (1+ε)r(L′) intersects every element of L. The running time of the algorithm is O(n^{Δ+1} dpoly (Δ/ε)). For the case of lines or line segments (Δ=1), the (expected) running time of the algorithm can be improved to O(ndpoly (1/ε)). We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space.

Original language | English |
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Pages (from-to) | 537-560 |

Number of pages | 24 |

Journal | Discrete and Computational Geometry |

Volume | 40 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2008 |

### Bibliographical note

Funding Information:Research of L.J. Schulman supported in part by an NSF ITR and the Okawa Foundation.

Funding Information:

Work was done when M. Langberg was a postdoctoral scholar at the California Institute of Technology. Research supported in part by NSF grant CCF-0346991.

## Keywords

- Approximation
- Clustering
- Core set
- Helly theorem
- Incomplete data
- Inference
- k-center