Multivariate Regression Depth

TL;DR

This paper generalizes the concept of regression depth from hyperplanes to k-flats in R^d, proving the existence of deep flats and providing an efficient approximation algorithm for finding the deepest flat.

Contribution

It introduces a generalized notion of regression depth for k-flats and proves the existence of deep flats for any k and d, along with a linear-time approximation algorithm.

Findings

Deep k-flats always exist with a constant fraction of points

A linear-time (1+epsilon)-approximation algorithm is developed

The generalization includes the classical center points as a special case

Abstract

The regression depth of a hyperplane with respect to a set of n points in R^d is the minimum number of points the hyperplane must pass through in a rotation to vertical. We generalize hyperplane regression depth to k-flats for any k between 0 and d-1. The k=0 case gives the classical notion of center points. We prove that for any k and d, deep k-flats exist, that is, for any set of n points there always exists a k-flat with depth at least a constant fraction of n. As a consequence, we derive a linear-time (1+epsilon)-approximation algorithm for the deepest flat.