Applications of Group Theory to Representation for Computational Intelligence

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.