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一种广泛使用的测度。然而，在高维空间中，Tukey深度的精确计算是一个公认的难题。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何时能够返回Tukey深度的良好近似。我们研究了数据服从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和趋近于零的深度；而对于任意中等深度点，任何良好近似都需要指数级复杂度。