A (Hilbert) geometric algorithm for approximating the halfspace depth of a point in a convex body

Halfspace (or Tukey) depth is a fundamental and robust measure of centrality of data points in multivariate datasets. Computing the depth of a point with respect to the uniform distribution on an open convex body in $\mathbb{R}^d$ is a natural algorithmic problem. While the coarser task of testing membership in convex bodies has been extensively studied, the refined problem of evaluating depth has received comparatively little attention in the literature. In this work, we present an algorithm for approximating the halfspace depth of a point in an open convex body $K \subset \mathbb{R}^d.$ To the best of our knowledge, this is the first deterministic algorithm for this problem. As part of our approach, we design an algorithm for answering approximate membership queries for the depth-trimmed regions of $K$ (i.e., the superlevel sets of the depth function). Our data structure is inspired by recent work of Abdelkader and Mount [SOSA 2024], wherein approximate membership queries for $K$ are answered using geometric structures derived from the Hilbert metric on $K.$ A key component underlying our data structure is a novel quantitative comparison between the depth-trimmed regions and the Hilbert metric balls of $K$. Lastly, to highlight the computational expense of the problem, we present an algorithm for determining the exact depth of a point in an open planar convex polygon presented as the intersection of finitely many halfplanes.