This paper studies how to generalize Tukey's depth to problems defined in a restricted space that may be curved or have boundaries, and to problems with a nondifferentiable objective. First, using a manifold approach, we propose a broad class of Riemannian depth for smooth problems defined on a Riemannian manifold, and showcase its applications in spherical data analysis, principal component analysis, and multivariate orthogonal regression. Moreover, for nonsmooth problems, we introduce additional slack variables and inequality constraints to define a novel slacked data depth, which can perform center-outward rankings of estimators arising from sparse learning and reduced rank regression. Real data examples illustrate the usefulness of some proposed data depths.

翻译：本文研究如何将Tukey深度推广至定义在可能存在弯曲或边界限制空间中的问题，以及具有不可微目标函数的问题。首先，采用流形方法，我们针对定义在黎曼流形上的光滑问题提出了一类广义的黎曼深度，并展示了其在球面数据分析、主成分分析和多元正交回归中的应用。此外，针对非光滑问题，我们引入附加松弛变量和不等式约束，定义了一种新颖的松弛数据深度，可对稀疏学习和降秩回归产生的估计量进行中心向外排序。真实数据实例验证了所提出部分数据深度的实用性。