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的深度概括到限制空间中可能弯曲或有边界的问题,以及非差别目标的问题。 首先,我们采用多种方法,提出一个广泛的里曼尼深度类别,用于平滑里曼多元体上界定的问题,并展示其在球体数据分析、主要组成部分分析和多变量正方形回归方面的应用。 此外,对于非单向问题,我们引入了额外的松懈变量和不平等制约,以界定新的松懈的数据深度,这可以对因稀疏学习和低级回归而形成的测算员进行中向外排序。真实数据实例说明了某些拟议数据深度的有用性。