A new approach for the computation of halfspace depth in high dimensions

Abstract

ABSTRACTHalfspace depth (HD), aka Tukey depth, is one of the most prevailing depth notions among all its competitors. To exactly compute the HD in is a challenging task nevertheless, due to its definition involving infinitely many directional projections. Existing algorithms to compute HD in (d ⩾ 2), more or less involve data projection to the directions perpendicular to hyperplanes (therefore involving polytopes or polyhedral cones) are either dimension d-, or sample size n-limited, or slow. Thanks to the work of Merkle and Bogićević and Merkle, algorithms for the fast computation of HD in ultra high dimension d for large n are proposed and addressed. They are sheer ball-based (or essentially point-wise distances) computations. The worst time complexity to approximately compute the depth m/n of a single point is O((d + m + log (n))n2) which seems to be one of the best known results so far among the competitors for extra large d and n. Unlike most previous algorithms, they do not require the underlying dataset to be in general position. As by-products of the algorithm, some depth regions (or depth level sets), and depth of sample points also could be obtained during the depth calculation of a given point.