|Title||Regression Depth and Center Points
|in||Discrete and Computational Geometry|
Nina Amenta, Marshall Bern, David Eppstein, Shang-Hua Teng |
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.