Regression Depth and Center Points
<|reference_start|>Regression Depth and Center Points: We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(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 ceiling(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.<|reference_end|>
@article{amenta1998regression,
    title={Regression Depth and Center Points},
    author={Nina Amenta, Marshall Bern, David Eppstein, Shang-Hua Teng},
    journal={Discrete Comput. Geom. 23(3):305-323, 2000},
    year={1998},
    doi={10.1007/PL00009502},
    archivePrefix={arXiv},
    eprint={cs/9809037},
    primaryClass={cs.CG math.CO}
}