# Developing a Foundation of Vector Symbolic Architectures Using Category Theory

arXiv: https://arxiv.org/abs/2501.05368 · HTML: https://mlsanity.com/paper/2501.05368/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 53 references that are in the map.
- First posted on arXiv 9 Jan 2025.

## Builds on (most cited first, 9 of 9)

- [A comparison of Vector Symbolic Architectures](https://mlsanity.com/paper/2001.11797/) (2020)
- [Categorical Foundations of Gradient-Based Learning](https://mlsanity.com/paper/2103.01931/) (2021)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [Category Theory in Machine Learning](https://mlsanity.com/paper/2106.07032/) (2021)
- [Variable Binding for Sparse Distributed Representations: Theory and Applications](https://mlsanity.com/paper/2009.06734/) (2020)
- [Representing Objects, Relations, and Sequences](https://mlsanity.com/paper/1501.07627/) (2015)
- [Robust computation with rhythmic spike patterns](https://mlsanity.com/paper/1901.07718/) (2019)
- [Hyperdimensional Computing with Spiking-Phasor Neurons](https://mlsanity.com/paper/2303.00066/) (2023)
- [Should Semantic Vector Composition be Explicit? Can it be Linear?](https://mlsanity.com/paper/2104.06555/) (2021)

## Abstract (arXiv)

Connectionist approaches to machine learning, i.e. neural networks, are enjoying a considerable 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 generalise from vectors to co-presheaves, and describe VSA operations as the right Kan extensions of the external tensor product. This formalisation involves a proof that the right Kan extension in such cases can be expressed as simple, element-wise operations. We validate our formalisation with worked examples that connect to current VSA implementations, while suggesting new possible designs for VSAs.