# Optimal quadratic binding for relational reasoning in vector symbolic neural architectures

arXiv: https://arxiv.org/abs/2204.07186 · HTML: https://mlsanity.com/paper/2204.07186/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 45 references that are in the map.
- First posted on arXiv 14 Apr 2022.

## Builds on (most cited first, 13 of 13)

- [VQA: Visual Question Answering](https://mlsanity.com/paper/1505.00468/) (2015)
- [CLEVR: A Diagnostic Dataset for Compositional Language and Elementary Visual Reasoning](https://mlsanity.com/paper/1612.06890/) (2016)
- [A simple neural network module for relational reasoning](https://mlsanity.com/paper/1706.01427/) (2017)
- [A Review of Relational Machine Learning for Knowledge Graphs](https://mlsanity.com/paper/1503.00759/) (2015)
- [On the Binding Problem in Artificial Neural Networks](https://mlsanity.com/paper/2012.05208/) (2020)
- [Holographic Embeddings of Knowledge Graphs](https://mlsanity.com/paper/1510.04935/) (2015)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [Tips and Tricks for Visual Question Answering: Learnings from the 2017 Challenge](https://mlsanity.com/paper/1708.02711/) (2017)
- [A comparison of Vector Symbolic Architectures](https://mlsanity.com/paper/2001.11797/) (2020)
- [A theory of sequence indexing and working memory in recurrent neural networks](https://mlsanity.com/paper/1803.00412/) (2018)
- [Variable Binding for Sparse Distributed Representations: Theory and Applications](https://mlsanity.com/paper/2009.06734/) (2020)
- [Learning with Holographic Reduced Representations](https://mlsanity.com/paper/2109.02157/) (2021)
- [Representing Objects, Relations, and Sequences](https://mlsanity.com/paper/1501.07627/) (2015)

## Built on by (most cited first, 2 of 2)

- [The Emergent Symbolic Structure of Artificial Neural Networks](https://mlsanity.com/paper/2608.29530/) (2026)
- [Efficient Vector Symbolic Architectures from Histogram Recovery](https://mlsanity.com/paper/2511.01838/) (2025)

## Abstract (arXiv)

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous works introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs as well as the octonion and previously-proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.