Analysis of incomplete data and an intrinsic-dimension Helly theorem

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′ ⊆ ℒsuch 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 ²), 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+1d poly(1/ε)). For the case of lines or line segments (Δ = 1), the (expected) running time of the algorithm can be improved to O(nd poly(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.