Smoothed Analysis of the Expected Number of Maximal Points in Two Dimensions

TL;DR

This paper analyzes how the expected number of maximal points in a two-dimensional set changes under random noise perturbations of the underlying distribution, extending classical results beyond uniform distributions.

Contribution

It provides a comprehensive analysis of the expected number of maxima when points are sampled from perturbed distributions, including various Lp-ball convolutions, for any noise level delta.

Findings

Expected maxima grow slowly with n under small noise

Distribution perturbations significantly affect the number of maxima

Results extend classical uniform distribution analyses

Abstract

The {\em Maximal} points in a set S are those that aren't {\em dominated} by any other point in S. Such points arise in multiple application settings in which they are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Because of their ubiquity, there is a large literature on the {\em expected} number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This work was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let Ballp denote the uniform distribution from the 2-d unit Lp ball, delta Ballq denote the 2-d Lq-ball, of radius delta and Ballpq be the convolution of the two distributions, i.e., a point v in Ballp is reported with an error chosen from delta Ballq. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,\delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type Ballpq where p,q in {1,2,infty} for delta > 0 and also of the type Ballp infty-q, where q in [1,infty) for delta > 0.