The Most Dispersed Subset of Random Points in $\mathbb{R}^d$

TL;DR

This paper analytically characterizes the maximum dispersion achievable among selected individuals with traits in high-dimensional space, using mean-field and replica methods, and identifies the optimal subset structure.

Contribution

It develops two analytical approaches to compute the full statistics of maximal dispersion in high dimensions and characterizes the optimal subset as points outside a self-consistently determined ball.

Findings

Optimal subset consists of points outside a certain radius in high dimensions.

Analytical formulas match numerical simulations and heuristic algorithms.

The methods apply to traits with rotationally symmetric distributions.

Abstract

Consider a population of $N$ individuals, each having $d\geq 1$ different traits, and an additive measure, called dispersion, which rewards large pairwise separations between traits. The goal is to select $M\leq N$ individuals such that their traits are as dispersed as possible. We compute analytically the full statistics (including large deviation tails) of the maximally achievable dispersion among sub-populations of size $M$ when the traits are independent and identically distributed. Two complementary approaches are developed, one based on a mean-field theory for order statistics, and the other on the replica method from the field of disordered systems. In all dimensions $d$, and for rotationally symmetric distributions, the optimal subset for large populations consists of all points lying outside a $d$-dimensional ball whose radius is determined self-consistently. For a single trait ($d=1$), the statistics of the maximal dispersion can be tackled for finite $N,M$ as well. The formulae we obtained are corroborated by numerical simulations on small instances and by heuristic algorithms that find near-optimal solutions.