Lower bounds for k-distance approximation

TL;DR

This paper proves that the halving polyhedron formed from random points on a sphere approximates the unit ball as the number of points increases, providing lower bounds on the complexity of distance approximation methods.

Contribution

It establishes probabilistic lower bounds for the complexity of k-distance approximation based on geometric properties of random point sets on spheres.

Findings

Halving polyhedron approximates the unit ball with high probability

Lower bounds on approximation complexity grow with Omega(d log d)

Results hold for large random point sets on spheres

Abstract

Consider a set P of N random points on the unit sphere of dimension $d-1$, and the symmetrized set S = P union (-P). The halving polyhedron of S is defined as the convex hull of the set of centroids of N distinct points in S. We prove that after appropriate rescaling this halving polyhedron is Hausdorff close to the unit ball with high probability, as soon as the number of points grows like $Omega(d log(d))$. From this result, we deduce probabilistic lower bounds on the complexity of approximations of the distance to the empirical measure on the point set by distance-like functions.