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

TL;DR

This paper extends Tukey's depth to curved spaces and nonsmooth problems, introducing Riemannian and slacked data depths, with applications in spherical data, PCA, and regression.

Contribution

It proposes a broad class of Riemannian depths for smooth manifold problems and a novel slacked data depth for nonsmooth problems, enhancing data ranking methods.

Findings

Demonstrates applications in spherical data analysis and PCA.

Introduces a new depth for nonsmooth optimization problems.

Provides real data examples validating the methods.

Abstract

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.