Exploring Symmetry and Representation in Vector Symbolic Architectures

In this thesis, I explore and extend the methods of Vector Symbolic Architectures (VSAs), with a particular focus on Holographic Reduced Representations (HRRs) and group VSAs.Through a consideration of abstract groups and linear representation theory, I provide detailed analyses of these systems to clarify their underlying algebraic structures, and suggest a unifying framework that enables the construction of new HRR-like VSA systems.In addition I present novel VSA models, including a memory retrieval model based on Hopfield networks for use with unitary HRRs and other group VSA generalizations, and a resonator network model that solves for rotations in 3D space.I also introduce unitary group VSAs, and show that such VSAs satisfy the notion of a maximally expressive VSA system under certain formal assumptions, enabling the representation of a diverse set of structured data relationships.i Although I am the sole author of this work, the process that culminated in this thesis involved many people who in their own ways contributed to the words on these pages.Without their support, the completion of this thesis would not have been possible.Below, I acknowledge some of these influential people, although there are many others to whom I owe my gratitude.Thank you first and foremost to my thesis supervisor Dr. Robert L. West.Your continued encouragement to follow my curiosity and to ask strange questions about strange topics helped me find the deeper connections I sought, not only concerning the topics discussed here, but also those related to cognition, artificial intelligence, psychology, narrative, spirituality, and about life in general.All the time spent with you talking in coffee shops, on walks in the park and by the canal, and even during aimless strolls through neighborhoods was immensely rewarding, and I am grateful to you for all the experiences we shared.Thank you to my wife Katie and to my mom Ayten for your endless love, patience, and continued support in many ways over the last few years of my academic journey.My family is truly my foundation-I cannot fully express here the level of appreciation I have for what you have done for me to help me accomplish my goals.I love you both so much.I must also thank my thesis committee members Dr. Mary A. Kelly and Dr. Angelo B. Mingarelli.Your feedback of my work was essential in reaching the finish line with a more polished thesis document, and our discussions helped me better understand the larger context of my work, especially regarding how it could be communicated to a general academic audience.I am also grateful to Dr. Kelly for exposing me to Vector Symbolic Architectures in the first place, and for giving me the opportunity to collaborate with you and to discuss exciting research.Last but not least, I want to

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