# Approximate Calculation of Tukey's Depth and Median With High-dimensional Data

arXiv: https://arxiv.org/abs/1812.03174 · HTML: https://mlsanity.com/paper/1812.03174/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [Tukey Depths and Hamilton-Jacobi Differential Equations](https://mlsanity.com/paper/2104.01648/) (2021)

## Abstract (arXiv)

We present a new fast approximate algorithm for Tukey (halfspace) depth level sets and its implementation-ABCDepth. Given a d-dimensional data set for any d≥ 1, the algorithm is based on a representation of level sets as intersections of balls in ℝd. Our approach does not need calculations of projections of sample points to directions. This novel idea enables calculations of approximate level sets in very high dimensions with complexity which is linear in d, which provides a great advantage over all other approximate algorithms. Using different versions of this algorithm we demonstrate approximate calculations of the deepest set of points ("Tukey median") and Tukey's depth of a sample point or out-of-sample point, all with a linear in d complexity. An additional theoretical advantage of this approach is that the data points are not assumed to be in "general position". Examples with real and synthetic data show that the executing time of the algorithm in all mentioned versions in high dimensions is much smaller than the time of other implemented algorithms. Also, our algorithms can be used with thousands of multidimensional observations.