On the quality of randomized approximations of Tukey's depth

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.

@article{briend2025_2309.05657,
  title={ On the quality of randomized approximations of Tukey's depth },
  author={ Simon Briend and Gábor Lugosi and Roberto Imbuzeiro Oliveira },
  journal={arXiv preprint arXiv:2309.05657},
  year={ 2025 }
}