Tukey Depths and Hamilton-Jacobi Differential Equations

TL;DR

This paper links the Tukey depth, a robust statistical measure, to Hamilton-Jacobi PDEs, establishing existence, uniqueness, and bounds of solutions, and explores numerical methods for solving these equations.

Contribution

It introduces a novel PDE framework for analyzing the Tukey depth, including proofs of solution properties and connections to viscosity solutions.

Findings

The associated PDE has a unique viscosity solution.

The viscosity solution bounds the Tukey depth from below.

In some cases, the Tukey depth equals the viscosity solution.

Abstract

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.