Tukey's depth offers a powerful tool for nonparametric inference and estimation, but also encounters serious computational and methodological difficulties in modern statistical data analysis. This paper studies how to generalize and compute Tukey-type depths in multi-dimensions. A general framework of influence-driven polished subspace depth, which emphasizes the importance of the underlying influence space and discrepancy measure, is introduced. The new matrix formulation enables us to utilize state-of-the-art optimization techniques to develop scalable algorithms with implementation ease and guaranteed fast convergence. In particular, half-space depth as well as regression depth can now be computed much faster than previously possible, with the support from extensive experiments. A companion paper is also offered to the reader in the same issue of this journal.

翻译：Tukey深度为非参数推断与估计提供了有力工具，但在现代统计数据分析中面临严重的计算与方法论困难。本文研究如何在多维空间中泛化并计算Tukey型深度。我们提出了一种基于影响驱动的抛光子空间深度通用框架，该框架强调了底层影响空间与差异度量的重要性。新的矩阵化表述使我们能够利用最先进的优化技术，开发出易于实现且保证快速收敛的可扩展算法。特别地，半空间深度与回归深度如今能以远超以往的速度进行计算，大量实验为这一结论提供了支持。读者可在同一期期刊中查阅本文的姊妹篇。