A Mathematical Unification of Geometric Crossovers Defined on Phenotype Space

Geometric crossover is a representation-independent definition of crossover based on the distance of the search space interpreted as a metric space. It generalizes the traditional crossover for binary strings and other important recombination operators for the most frequently used representations. Using a distance tailored to the problem at hand, the abstract definition of crossover can be used to design new problem specific crossovers that embed problem knowledge in the search. This paper is motivated by the fact that genotype-phenotype mapping can be theoretically interpreted using the concept of quotient space in mathematics. In this paper, we study a metric transformation, the quotient metric space, that gives rise to the notion of quotient geometric crossover. This turns out to be a very versatile notion. We give many example applications of the quotient geometric crossover.

💡 Research Summary

The paper presents a mathematically rigorous unification of geometric crossover operators by shifting the focus from genotype‑centric representations to phenotype‑centric spaces. Geometric crossover, originally defined as a distance‑based recombination in a metric space, abstracts traditional crossover mechanisms for binary strings, permutations, trees, and other encodings. While this abstraction is representation‑independent, it ignores the fact that many distinct genotypes map to the same phenotype, leading to redundant exploration and inefficiency.

To address this, the authors invoke the concept of a quotient space from topology. Given a genotype set X, a phenotype set Y, and a mapping φ: X → Y, an equivalence relation ~ is induced on X where x₁ ~ x₂ iff φ(x₁) = φ(x₂). The quotient set X/∼ consists of equivalence classes, each representing a unique phenotype. A quotient metric d_Q is defined on X/∼ by taking the minimum distance between any representatives of two classes: d_Q(