An application of the universality theorem for Tverberg partitions to data depth and hitting convex sets

Abstract

We show that, as a consequence of a new result of Pór on universal Tverberg partitions, any large-enough set P of points in R d has a (d + 2)-sized subset whose Radon point has half-space depth at least c d • |P |, where c d ∈ (0, 1) depends only on d. We then give an application of this result to computing weak -nets by random sampling. We further show that given any set P of points in R d and a parameter > 0, there exists a set of Od 2 -dimensional simplices (ignoring polylogarithmic factors) spanned by points of P such that they form a transversal for all convex objects containing at least • |P | points of P .

On the robustness to adversarial corruption and to heavy-tailed data of the Stahel-Donoho median of means

“…There are several ways to measure how 'deep' is a vector with respect to a cloud of points, see for instance the half-space depth of Tukey [64,58], the simplicial depth [44,46], Mahalanobis depth or the projection depth [45]. Taking a point with maximal depth is usually seen as a way to define a median in R d (see Radon points [2] or Fermat Points [27]). There are therefore several ways to define a median of a cloud of points in R d .…”