Statistics of Partial Minima

TL;DR

This paper investigates the statistical properties of partial minima in high-dimensional spaces, revealing algebraic decay and logarithmic growth patterns depending on the dominance parameter k and dimension d.

Contribution

It introduces exact probabilistic methods and heuristic scaling techniques to analyze the distribution and count of partial minima in multi-objective optimization.

Findings

Average number of partial minima decays algebraically with N for 1<=k<d.

Largest coordinate distributions follow algebraic decay with specific exponents.

Number of minima grows logarithmically with N when k=d.

Abstract

Motivated by multi-objective optimization, we study extrema of a set of N points independently distributed inside the d-dimensional hypercube. A point in this set is k-dominated by another point when at least k of its coordinates are larger, and is a k-minimum if it is not k-dominated by any other point. We obtain statistical properties of these partial minima using exact probabilistic methods and heuristic scaling techniques. The average number of partial minima, A, decays algebraically with the total number of points, A ~ N^{-(d-k)/k}, when 1<=k<d. Interestingly, there are k-1 distinct scaling laws characterizing the largest coordinates as the distribution P(y_j) of the jth largest coordinate, y_j, decays algebraically, P(y_j) ~ (y_j)^{-alpha_j-1}, with alpha_j=j(d-k)/(k-j) for 1<=j<=k-1. The average number of partial minima grows logarithmically, A ~ [1/(d-1)!](ln N)^{d-1}, when k=d. The full distribution of the number of minima is obtained in closed form in two-dimensions.