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Geometric landscape of homologous crossover for syntactic trees
 In Proceedings of CEC 2005
, 2005
"... Abstract Geometric crossover and geometric mutation are representationindependent operators that are welldefined once a notion of distance over the solution space is defined. They were obtained as generalizations of genetic operators for binary strings and real vectors. Our geometric framework has ..."
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Cited by 14 (13 self)
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Abstract Geometric crossover and geometric mutation are representationindependent operators that are welldefined once a notion of distance over the solution space is defined. They were obtained as generalizations of genetic operators for binary strings and real vectors. Our geometric framework has been successfully applied to the permutation representation leading to a clarification and a natural unification of this domain. The relationship between search space, distances and genetic operators for syntactic trees is little understood. In this paper we apply the geometric framework to the syntactic tree representation and show how the wellknown structural distance is naturally associated with homologous crossover and subtree mutation. 1
Product geometric crossover
 in Proceedings of the 9th International Conference on Parallel Problem Solving from Nature (PPSN ’06
, 2006
"... Abstract. Geometric crossover is a representationindependent 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 represen ..."
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Cited by 9 (8 self)
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Abstract. Geometric crossover is a representationindependent 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. In this paper, we introduce the important notion of product geometric crossover that allows to build new geometric crossovers combining preexisting geometric crossovers in a simple way. 1
Geometric Crossover for Sets, Multisets and Partitions
 In Proceedings of the Parallel Problem Solving from Nature Conference
, 2006
"... Abstract. This paper extends a geometric framework for interpreting crossover and mutation [5] to the case of sets and related representations. We show that a deep geometric duality exists between the set representation and the vector representation. This duality reveals the equivalence of geometric ..."
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Cited by 7 (7 self)
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Abstract. This paper extends a geometric framework for interpreting crossover and mutation [5] to the case of sets and related representations. We show that a deep geometric duality exists between the set representation and the vector representation. This duality reveals the equivalence of geometric crossovers for these representations. 1
Inbreeding properties of geometric crossover and nongeometric recombinations
 In Proceedings of the Foundations of Genetic Algorithms
, 2007
"... Abstract. Geometric crossover is a representationindependent generalization of traditional crossover for binary strings. It is defined using the distance associated to the search space in a simple geometric way. Many interesting recombination operators for the most frequently used representations a ..."
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Cited by 5 (4 self)
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Abstract. Geometric crossover is a representationindependent generalization of traditional crossover for binary strings. It is defined using the distance associated to the search space in a simple geometric way. Many interesting recombination operators for the most frequently used representations are geometric crossovers under some suitable distance. Being a geometric crossover is useful because there is a growing number of theoretical results that apply to this class of operators. To show that a given recombination operator is a geometric crossover, it is sufficient to find a distance for which offspring are in the metric segment between parents associated with this distance. However, proving that a recombination operator is not a geometric crossover requires to prove that such an operator is not a geometric crossover under any distance. In this paper we develop some theoretical tools to prove nongeometricity results and show that some wellknown operators are not geometric. 1
Geometric Crossover for Supervised Motif Discovery
"... Motif discovery is a general and important problem in bioinformatics, as motifs often are used to infer biologically important sites in biomolecular sequences. Many problems in bioinformatics are naturally cast in terms of sequences, and distance measures for sequences derived from edit distance is ..."
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Motif discovery is a general and important problem in bioinformatics, as motifs often are used to infer biologically important sites in biomolecular sequences. Many problems in bioinformatics are naturally cast in terms of sequences, and distance measures for sequences derived from edit distance is fundamental in bioinformatics. Geometric Crossover is a representationindependent definition of crossover based on a distance on the solution space. Using a distance measure that is tailored to the problem at hand allows the design of crossovers that embed problem knowledge in the search. In this paper we apply this theoretically motivated operator to motif discovery in protein sequences and report encouraging experimental results.
Geometric Crossover for Supervised Motif Discovery
, 2006
"... Motif discovery is a general and important problem in bioinformatics, as motifs often are used to infer biologically important sites in biomolecular sequences. Many problems in bioinformatics are naturally cast in terms of sequences, and distance measures for sequences derived from edit distance is ..."
Abstract
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Motif discovery is a general and important problem in bioinformatics, as motifs often are used to infer biologically important sites in biomolecular sequences. Many problems in bioinformatics are naturally cast in terms of sequences, and distance measures for sequences derived from edit distance is fundamental in bioinformatics. Geometric Crossover is a representationindependent definition of crossover based on a distance on the solution space. Using a distance measure that is tailored to the problem at hand allows the design of crossovers that embed problem knowledge in the search. In this paper we apply this theoretically motivated operator to motif discovery in protein sequences and report encouraging experimental results.
Quotient Geometric Crossovers and Redundant Encodings
, 2011
"... We extend a geometric framework for the interpretation of search operators to encompass the genotypephenotype mapping derived from an equivalence relation defined by an isometry group. We show that this mapping can be naturally interpreted using the concept of quotient space, in which the original ..."
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We extend a geometric framework for the interpretation of search operators to encompass the genotypephenotype mapping derived from an equivalence relation defined by an isometry group. We show that this mapping can be naturally interpreted using the concept of quotient space, in which the original space corresponds to the genotype space and the quotient space corresponds to the phenotype space. Using this characterization, it is possible to define induced geometric crossovers on the phenotype space (called quotient geometric crossovers). These crossovers have very appealing properties for nonsynonymously redundant encodings, such as reducing the size of the search space actually searched, removing the low locality from the encodings, and allowing a more informed search by utilizing distances better tailored to the specific solution interpretation. Interestingly, quotient geometric crossovers act on genotypes but have an effect equivalent to geometric crossovers acting directly on the phenotype space. This property allows us to actually implement them even when phenotypes cannot be represented directly. We give four example applications of quotient geometric crossovers for nonsynonymously redundant encodings and demonstrate their superiority experimentally.
Mathematical Interpretation between Genotype and Phenotype Spaces and Induced Geometric Crossovers
, 2009
"... In this paper, we present that genotypephenotype mapping can be theoretically interpreted using the concept of quotient space in mathematics. Quotient space can be considered as mathematicallydefined phenotype space in the evolutionary computation theory. The quotient geometric crossover has the e ..."
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In this paper, we present that genotypephenotype mapping can be theoretically interpreted using the concept of quotient space in mathematics. Quotient space can be considered as mathematicallydefined phenotype space in the evolutionary computation theory. The quotient geometric crossover has the effect of reducing the search space actually searched by geometric crossover, and it introduces problem knowledge in the search by using a distance better tailored to the specific solution interpretation. Quotient geometric crossovers are directly applied to the genotype space but they have the effect of the crossovers performed on phenotype space. We give many example applications of the quotient geometric crossover.
A Mathematical Unification of Geometric Crossovers Defined on Phenotype Space
, 2008
"... Geometric crossover is a representationindependent 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 definition of crossover ..."
Abstract
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Geometric crossover is a representationindependent 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 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 genotypephenotype 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.