Regression depth, introduced by Rousseeuw and Hubert in 1999, is a notion that measures how good of a regression hyperplane a given query hyperplane is with respect to a set of data points. Under projective duality, this can be interpreted as a depth measure for query points with respect to an arrangement of data hyperplanes. The study of depth measures for query points with respect to a set of data points has a long history, and many such depth measures have natural counterparts in the setting of hyperplane arrangements. For example, regression depth is the counterpart of Tukey depth. Motivated by this, we study general families of depth measures for hyperplane arrangements and show that all of them must have a deep point. Along the way we prove a Tverberg-type theorem for hyperplane arrangements, giving a positive answer to a conjecture by Rousseeuw and Hubert from 1999. We also get three new proofs of the centerpoint theorem for regression depth, all of which are either stronger or more general than the original proof by Amenta, Bern, Eppstein, and Teng. Finally, we prove a version of the center transversal theorem for regression depth.

翻译：回归深度由Rousseeuw和Hubert于1999年提出，是一种衡量给定查询超平面相对于一组数据点作为回归超平面优劣程度的概念。在射影对偶性下，这可被解释为查询点相对于数据超平面排列的深度度量。针对数据点集中查询点的深度度量研究历史悠久，且许多此类度量在超平面排列场景中具有自然对应物。例如，回归深度即Tukey深度的对应物。受此启发，我们研究了超平面排列的深度度量一般族，并证明所有此类度量必然存在深度点。在此过程中，我们证明了超平面排列的Tverberg型定理，对Rousseeuw与Hubert于1999年提出的猜想给出了肯定回答。我们还获得了回归深度中心点定理的三个新证明，这些证明均比Amenta、Bern、Eppstein和Teng的原始证明更加强大或更具一般性。最后，我们证明了回归深度的中心横截定理版本。