On the quality of randomized approximations of Tukey's depth

Simon Briend; Gábor Lugosi; Roberto Imbuzeiro Oliveira

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arXiv · 2309.05657

On the quality of randomized approximations of Tukey's depth

Simon Briend·Gábor Lugosi·Roberto Imbuzeiro Oliveira

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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.

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Simon Briend, Gábor Lugosi, Roberto Imbuzeiro Oliveira. 2023-09-11. On the quality of randomized approximations of Tukey's depth. https://arxiv.org/abs/2309.05657

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