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Regression Depth and Center Points

  • Author(s): Amenta, Nina;
  • Bern, Marshall;
  • Eppstein, David;
  • Teng, Shang-Hua
  • et al.
Abstract

We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least [n/(d+1)], as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least [n/(d+1)]hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.

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