Uniform convergence rates for the approximated halfspace and projection depth

TL;DR

This paper provides a theoretical analysis of the approximation of halfspace and projection depths using randomized projections, establishing uniform convergence rates and guidelines for choosing the number of projections.

Contribution

It offers the first theoretical background and sharp convergence rates for the randomized approximation of these depths, guiding practical implementation.

Findings

Established uniform consistency results for depth approximations.

Derived sharp convergence rates for elliptically symmetric distributions.

Provided guidelines for selecting the number of random projections.

Abstract

The computational complexity of some depths that satisfy the projection property, such as the halfspace depth or the projection depth, is known to be high, especially for data of higher dimensionality. In such scenarios, the exact depth is frequently approximated using a randomized approach: The data are projected into a finite number of directions uniformly distributed on the unit sphere, and the minimal depth of these univariate projections is used to approximate the true depth. We provide a theoretical background for this approximation procedure. Several uniform consistency results are established, and the corresponding uniform convergence rates are provided. For elliptically symmetric distributions and the halfspace depth it is shown that the obtained uniform convergence rates are sharp. In particular, guidelines for the choice of the number of random projections in order to achieve a given precision of the depths are stated.