Regression Depth and Center Points

TL;DR

This paper proves a conjecture about the existence of hyperplanes with high regression depth in any set of points, and explores related geometric and computational properties.

Contribution

It establishes the existence of hyperplanes with a minimum regression depth for any point set, confirming a conjecture by Rousseeuw and Hubert, and investigates related partitioning and complexity issues.

Findings

Existence of hyperplanes with regression depth at least ceiling(n/(d+1))

Dual result on points crossing hyperplanes

Analysis of partitioning and computational complexity of regression depth

Abstract

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.