# Tukey Depths and Hamilton-Jacobi Differential Equations

arXiv: https://arxiv.org/abs/2104.01648 · HTML: https://mlsanity.com/paper/2104.01648/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 48 references that are in the map.
- First posted on arXiv 4 Apr 2021.

## Builds on (most cited first, 5 of 5)

- [Exact computation of the halfspace depth](https://mlsanity.com/paper/1411.6927/) (2014)
- [The limit shape of convex hull peeling](https://mlsanity.com/paper/1805.08278/) (2018)
- [Data depth and floating body](https://mlsanity.com/paper/1809.10925/) (2018)
- [A Hamilton-Jacobi equation for the continuum limit of non-dominated sorting](https://mlsanity.com/paper/1302.5828/) (2013)
- [Approximate Calculation of Tukey's Depth and Median With High-dimensional Data](https://mlsanity.com/paper/1812.03174/) (2018)

## Built on by (most cited first, 3 of 3)

- [Hamilton-Jacobi equations on graphs with applications to semi-supervised learning and data depth](https://mlsanity.com/paper/2202.08789/) (2022)
- [Eikonal depth: an optimal control approach to statistical depths](https://mlsanity.com/paper/2201.05274/) (2022)
- [A Learned Generalized Geodesic Distance Function-Based Approach for Node Feature Augmentation on Graphs](https://mlsanity.com/paper/2407.01194/) (2024)

## Abstract (arXiv)

The widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical measure known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we derive the associated necessary conditions, which take the form of a first-order partial differential equation. We discuss the classical interpretation of this necessary condition as the viscosity solution of a Hamilton-Jacobi equation, but with a non-classical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases, we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges.