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.

翻译：Tukey深度（或称半空间深度）是衡量多元数据中心性的广泛使用的指标。然而，在高维空间中精确计算Tukey深度已知是一个困难问题。为此，研究者提出了Tukey深度的随机近似方法。本文探讨了此类随机算法在何种条件下能对Tukey深度给出良好近似。我们研究了数据采样自对数凹各向同性分布的情况。证明表明，若要求算法在维度上以多项式时间运行，则随机算法能正确近似最大深度$1/2$及接近零的深度。另一方面，对于任意中间深度点，任何良好近似都需要指数级复杂度。