On the quality of randomized approximations of Tukey's depth

TL;DR

This paper investigates the effectiveness of randomized algorithms in approximating Tukey's depth for high-dimensional data, showing they reliably estimate extreme depths but struggle with intermediate values unless exponential complexity is used.

Contribution

It provides theoretical bounds on when randomized algorithms can accurately approximate Tukey's depth in high dimensions, especially for data from log-concave isotropic distributions.

Findings

Randomized algorithms accurately approximate maximal and near-zero depths in polynomial time.

Approximating intermediate depths requires exponential complexity.

Results are specific to data sampled from log-concave isotropic distributions.

Abstract

Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's depth have been proposed. In this paper we explore when such randomized algorithms return a good approximation of Tukey's depth. We study the case when the data are sampled from a log-concave isotropic distribution. We prove that, if one requires that the algorithm runs in polynomial time in the dimension, the randomized algorithm correctly approximates the maximal depth $1/2$ and depths close to zero. On the other hand, for any point of intermediate depth, any good approximation requires exponential complexity.