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On the Quality of Randomized Approximations of Tukey's Depth
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DOI：10.1137/24M1654919.png)
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Abstract
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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.
Keywords:
Tukey depth
high-dimensional statistics
data analysis
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