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Applications of Group Theory to Representation for Computational Intelligence

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Title: Applications of Group Theory to Representation for Computational Intelligence
Author: Gilbert, Jeremy Alexander
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Ashlock, DanielPereira, Rajesh
Abstract: Representations Arising From Group Theory. This thesis introduces a novel approach to developing representations for evolutionary computation, using group theory as a foundation. The goal is to develop new representations which are better suited for navigating treacherous fitness landscapes, yielding improvements to algorithm performance over traditional methods. To construct such a representation, a selection of elements from a group are specified and used as generators to form a subgroup. The representation takes the form of words over the set of generators. An evolutionary algorithm is then able to search the space of words, which is a standard form of evolutionary algorithm. Multiple new representations are presented, built from additive vector groups, bijections of the unit interval, and affine transformations on Euclidean space. These representations can be used in a variety of applications, including real optimization, data normalization, image generation and modification, and point packing generation. Some can also be used to discretize a continuous search space, allowing the use of algorithms such as Monte Carlo Tree Search. The discrete nature of these representations also allows for use of a dictionary of previous optimal solutions. This permits an algorithm to find a diverse set of best fit solutions, by using the dictionary to exclude parts of the search space near solutions that have already been found, realized as prefixes of stored words. A parameter study is performed for each representation, and they are compared to conventional methods on a variety of test problems.
Date: 2022-01
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Related Publications: D. Ashlock and J. Gilbert. A discrete representation for real optimization with unique search properties. In Proceedings of the IEEE Symposium on the Foundations of Computational Intelligence, pages 54–61, 2014. DOI: 10.1109/FOCI.2014.7007807J. Gilbert and D. Ashlock. Evolvable warps for data normalization. In Proceedings of the IEEE 2016 Congress on Evolutionary Computation, pages 1562–1569, 2016. DOI: 10.1109/CEC.2016.7743975D. Ashlock and J. Gilbert. A greedy, generative, lattice representation for point packing. In Proceedings of the 2019 IEEE Congress on Evolutionary Computation, pages 3181– 3188, 2019. DOI: 10.1109/CEC.2019.8790058

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