Tukey Depths and Hamilton-Jacobi Differential Equations
This work addresses a fundamental statistical measure for robust machine learning, but it appears incremental as it extends existing PDE frameworks to Tukey depth.
The paper tackles the problem of computing Tukey depth in the continuum limit by deriving a first-order partial differential equation and proving it has a unique viscosity solution that bounds Tukey depth from below, with equality in some cases, and demonstrates numerical methods from optimal control.
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.