Statistical analysis with missing data
Statistical analysis with missing data
Approximation algorithms for projective clustering
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Approximate clustering via core-sets
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation Algorithms for k-Line Center
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
A (1 + ɛ)-approximation algorithm for 2-line-center
Computational Geometry: Theory and Applications
SIAM Journal on Computing
Convex Optimization
Approximate minimum enclosing balls in high dimensions using core-sets
Journal of Experimental Algorithmics (JEA)
On coresets for k-means and k-median clustering
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Approximating extent measures of points
Journal of the ACM (JACM)
How to get close to the median shape
Proceedings of the twenty-second annual symposium on Computational geometry
How to get close to the median shape
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Approximating largest convex hulls for imprecise points
Journal of Discrete Algorithms
Approximating largest convex hulls for imprecise points
WAOA'07 Proceedings of the 5th international conference on Approximation and online algorithms
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The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in Rd, 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(Δ4/ε2), 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(nΔ+1dpoly(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.