🤖 AI Summary
Exact computation of Tukey depth becomes intractable in high dimensions; this work investigates the fundamental feasibility limits of its randomized approximation. Method: Focusing on log-concave isotropic distributions, the authors employ tools from geometric probability, high-dimensional statistics, and random hyperplane sampling to rigorously analyze the computational complexity of estimating Tukey depth. Contribution/Results: They establish the first tight characterization: for points with depth near 0 or 1/2, polynomial-time randomized algorithms achieve efficient approximation; however, for any constant depth $c in (0,1/2)$, arbitrarily high-precision estimation requires exponential time. This reveals a sharp phase transition between depth value and computational complexity in Tukey depth approximation. The analysis precisely delineates the theoretical limits of randomized algorithms under the log-concave assumption, providing a foundational complexity-theoretic characterization and principled guidance for algorithm design in high-dimensional depth computation.
📝 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.
