论文标题
Eikonal深度：统计深度的最佳控制方法
Eikonal depth: an optimal control approach to statistical depths
论文作者
论文摘要
统计深度为较高维度的数据提供了对分位数和中位数的基本概括。本文提出了一种基于控制理论和Eikonal方程的新型全球定义的统计深度，该深度衡量了最小的概率密度，该密度必须在分布支持以外的点以外的指数中通过：例如空间无限。该深度易于解释和计算，表达捕获多模式的行为，并自然地扩展到非欧盟人的数据。我们证明了该深度的各种特性，并提供了计算考虑因素的讨论。特别是，我们证明了这种深度概念在异位限制的对抗模型下是强大的，这是Tukey深度不符合的属性。最后，我们在二维混合模型和MNIST的背景下给出了一些说明性示例。
Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.
