Tukey深度与Hamilton-雅可比微分方程
Tukey Depths and Hamilton-Jacobi Differential Equations
摘要：
现代机器学习的广泛应用增加了对鲁棒统计算法的需求。这项工作研究了一个这样的基本统计措施被称为Tukey深度。我们在连续统（人口）极限下研究这个问题。特别是，我们得到了相关的必要条件，它采取一阶偏微分方程的形式。我们讨论了这个必要条件的经典解释的黏性解的Hamilton-Jacobi方程，但与非经典的哈密顿与不连续的梯度在零。我们证明了该方程具有唯一的黏性解，并且该解总是从下面限制Tukey深度。在某些情况下，我们证明了Tukey深度等于黏性解，并给出了一些最优控制界直接处理偏微分方程的标准数值方法的例子。最后，我们概述了几个有前途的研究方向，无论是在新的数值算法和理论挑战。
英文摘要：
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.
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