Approximate computation of projection depths

TL;DR

This paper reviews and compares various approximate algorithms for computing projection depths in multivariate data, highlighting their efficiency and applicability in high-dimensional settings.

Contribution

It provides a comprehensive comparison of existing and new methods for approximating projection depths, addressing computational challenges in high dimensions.

Findings

Approximate algorithms offer practical solutions for high-dimensional data.

Comparison reveals trade-offs between accuracy and computational complexity.

New methods improve efficiency over existing approaches.

Abstract

Data depth is a concept in multivariate statistics that measures the centrality of a point in a given data cloud in $\IR^d$. If the depth of a point can be represented as the minimum of the depths with respect to all one-dimensional projections of the data, then the depth satisfies the so-called projection property. Such depths form an important class that includes many of the depths that have been proposed in literature. For depths that satisfy the projection property an approximate algorithm can easily be constructed since taking the minimum of the depths with respect to only a finite number of one-dimensional projections yields an upper bound for the depth with respect to the multivariate data. Such an algorithm is particularly useful if no exact algorithm exists or if the exact algorithm has a high computational complexity, as is the case with the halfspace depth or the projection depth. To compute these depths in high dimensions, the use of an approximate algorithm with better complexity is surely preferable. Instead of focusing on a single method we provide a comprehensive and fair comparison of several methods, both already described in the literature and original.