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

## TL;DR

This paper introduces a fast, approximate algorithm for calculating Tukey's depth and median in high-dimensional data, leveraging a novel intersection-of-balls approach that avoids projections and scales linearly with dimension.

## Contribution

The paper presents a new high-dimensional algorithm for Tukey depth and median calculation that is faster and more scalable than existing methods, without requiring data in general position.

## Key findings

- Algorithm is significantly faster in high dimensions.
- Complexity is linear in the dimension d.
- Can handle thousands of multidimensional observations.

## Abstract

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 $d\geq 1$, the algorithm is based on a representation of level sets as intersections of balls in $\mathbb{R}^d$. 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.