Fast implementation of the Tukey depth

TL;DR

This paper introduces two new combinatorial algorithms for efficiently computing the Tukey depth in multivariate data, significantly improving speed and memory usage over existing methods.

Contribution

It presents the first naive and a more efficient quasiconcavity-based algorithm for exact Tukey depth calculation in higher dimensions.

Findings

Algorithms run faster than existing methods.

Require minimal memory.

Compute exact Tukey depth.

Abstract

Tukey depth function is one of the most famous multivariate tools serving robust purposes. It is also very well known for its computability problems in dimensions $p \ge 3$. In this paper, we address this computing issue by presenting two combinatorial algorithms. The first is naive and calculates the Tukey depth of a single point with complexity $O\left(n^{p-1}\log(n)\right)$, while the second further utilizes the quasiconcave of the Tukey depth function and hence is more efficient than the first. Both require very minimal memory and run much faster than the existing ones. All experiments indicate that they compute the exact Tukey depth.