Developing a Foundation of Vector Symbolic Architectures Using Category Theory

At the risk of overstating the case, connectionist approaches to machine learning, i.e. neural networks, are enjoying a small vogue right now. However, these methods require large volumes of data and produce models that are uninterpretable to humans. An alternative framework that is compatible with neural networks and gradient-based learning, but explicitly models compositionality, is Vector Symbolic Architectures (VSAs). VSAs are a family of algebras on high-dimensional vector representations. They arose in cognitive science from the need to unify neural processing and the kind of symbolic reasoning that humans perform. While machine learning methods have benefited from category theoretical analyses, VSAs have not yet received similar treatment. In this paper, we present a first attempt at applying category theory to VSAs. Specifically, we conduct a brief literature survey demonstrating the lacking intersection of these two topics, provide a list of desiderata for VSAs, and propose that VSAs may be understood as a (division) rig in a category enriched over a monoid in Met (the category of Lawvere metric spaces). This final contribution suggests that VSAs may be generalised beyond current implementations. It is our hope that grounding VSAs in category theory will lead to more rigorous connections with other research, both within and beyond, learning and cognition.