Uniform convergence rates for the approximated halfspace and projection depth

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.

Approximate computation of projection depths

“…Our most important observation is a high performance of the optimization techniques (SA, RRS, CD, and NM) compared with the random methods (RS, RaSi). The two latter ones seem to (approximately) follow the bounds derived in Nagy, Dyckerhoff, and Mozharovskyi (2019) and are outperformed already before reaching 100 random directions. Further inspection shows that the improvement of simulated annealing (SA) is very weak, and minor improvement can be expected for even higher number of directions.…”

“…The convergence of min 1≤i≤N D( p i , z | p i , X ) for the halfspace depth and the projection depth has been extensively studied in Nagy, Dyckerhoff, and Mozharovskyi (2019). Given a precision , these results can be used to find N ( ) such that the error is approximately .…”