On Generalization and Computation of Tukey's Depth: Part I

TL;DR

This paper introduces a general framework and scalable algorithms for computing Tukey-type depths in multiple dimensions, significantly improving efficiency and applicability in modern statistical analysis.

Contribution

It proposes a novel influence-driven polished subspace depth framework and matrix formulation, enabling faster computation of half-space and regression depths.

Findings

Faster algorithms for Tukey's depth computation.

Effective use of optimization techniques for scalability.

Empirical validation demonstrating improved performance.

Abstract

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.