Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem
- 58 Downloads
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.
KeywordsClustering k-center Core set Incomplete data Helly theorem Approximation Inference
Unable to display preview. Download preview PDF.
- 1.Agarwal, P.K., Procopiuc, C.M.: Approximation algorithms for projective clustering. In: SODA’00: Proceedings of the Eleventh Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, 2000, pp. 538–547. SIAM, Philadelphia (2000) Google Scholar
- 2.Agarwal, P.K., Procopiuc, C.M., Varadarajan, K.R.: Approximation algorithms for k-line center. In: ESA’02: Proceedings of the 10th Annual European Symposium on Algorithms, London, UK, 2002, pp. 54–63. Springer, Berlin (2002) Google Scholar
- 5.Agarwal, P., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Current Trends in Combinatorial and Computational Geometry. Cambridge University Press, Cambridge (2005) Google Scholar
- 6.Bǎdoiu, M., Clarkson, K.L.: Smaller core-sets for balls. In: SODA’03: Proceedings of the Fourteenth Annual ACM–SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, 2003, pp. 801–802. SIAM, Philadelphia (2003) Google Scholar
- 8.Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2003) Google Scholar
- 9.Gao, J., Langberg, M., Schulman, L.: Clustering lines: classification of incomplete data. Manuscript (2006) Google Scholar
- 10.Har-Peled, S.: Private communication Google Scholar
- 16.Kumar, P., Mitchell, J.S.B., Yildirim, A.: Computing core-sets and approximate smallest enclosing hyperspheres in high dimensions. In: Proceedings of the Fifth Workshop on Algorithm Engineering and Experiments, pp. 45–55 (2003) Google Scholar
- 18.Little, R.J.A., Rubin, D.B.: Statistical Analysis with Missing Data. Wiley, New York (1986) Google Scholar