On the quality of randomized approximations of Tukey's depth
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866908435186974720 |
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| author | Briend, Simon Lugosi, Gábor Oliveira, Roberto Imbuzeiro |
| author_facet | Briend, Simon Lugosi, Gábor Oliveira, Roberto Imbuzeiro |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_05657 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the quality of randomized approximations of Tukey's depth Briend, Simon Lugosi, Gábor Oliveira, Roberto Imbuzeiro Machine Learning Probability 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. |
| title | On the quality of randomized approximations of Tukey's depth |
| topic | Machine Learning Probability |
| url | https://arxiv.org/abs/2309.05657 |