# Representing Objects, Relations, and Sequences

arXiv: https://arxiv.org/abs/1501.07627 · HTML: https://mlsanity.com/paper/1501.07627/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2015.

## Built on by (most cited first, 24 of 24)

- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [Vector Symbolic Architectures as a Computing Framework for Emerging Hardware](https://mlsanity.com/paper/2106.05268/) (2021)
- [Relational Composition in Neural Networks: A Survey and Call to Action](https://mlsanity.com/paper/2407.14662/) (2024)
- [Understanding Hyperdimensional Computing for Parallel Single-Pass Learning](https://mlsanity.com/paper/2202.04805/) (2022)
- [MIMONets: Multiple-Input-Multiple-Output Neural Networks Exploiting Computation in Superposition](https://mlsanity.com/paper/2312.02829/) (2023)
- [Hyperdimensional Feature Fusion for Out-Of-Distribution Detection](https://mlsanity.com/paper/2112.05341/) (2021)
- [Hyperdimensional Computing for Efficient Distributed Classification with Randomized Neural Networks](https://mlsanity.com/paper/2106.00881/) (2021)
- [Generalized Holographic Reduced Representations](https://mlsanity.com/paper/2405.09689/) (2024)
- [Shift-Equivariant Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2112.15475/) (2021)
- [Towards Generalization in Subitizing with Neuro-Symbolic Loss using Holographic Reduced Representations](https://mlsanity.com/paper/2312.15310/) (2023)
- [GPT-2 Through the Lens of Vector Symbolic Architectures](https://mlsanity.com/paper/2412.07947/) (2024)
- [High-dimensional distributed semantic spaces for utterances](https://mlsanity.com/paper/2104.00424/) (2021)
- [Recursive Binding for Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2201.11691/) (2022)
- [Optimal quadratic binding for relational reasoning in vector symbolic neural architectures](https://mlsanity.com/paper/2204.07186/) (2022)
- [DataWords: Getting Contrarian with Text, Structured Data and Explanations](https://mlsanity.com/paper/2111.05384/) (2021)
- [Orthogonal Matrices for MBAT Vector Symbolic Architectures, and a "Soft" VSA Representation for JSON](https://mlsanity.com/paper/2202.04771/) (2022)
- [Developing a Foundation of Vector Symbolic Architectures Using Category Theory](https://mlsanity.com/paper/2501.05368/) (2025)
- [Autonomous Learning with High-Dimensional Computing Architecture Similar to von Neumann's](https://mlsanity.com/paper/2503.23608/) (2025)
- ['Memory States' from Almost Nothing: Representing and Computing in a Non-associative Algebra](https://mlsanity.com/paper/2506.13768/) (2025)
- [A Vision Toward Energy-Efficient Domain-Specific Artificial Intelligence Models and Agents](https://mlsanity.com/paper/2510.22052/) (2025)
- [Semantic Multiplexing](https://mlsanity.com/paper/2511.13779/) (2025)
- [Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories](https://mlsanity.com/paper/2606.11391/) (2026)
- [A Unifying Perspective on Language Model Representations: From Filler-Role Structure to Mechanistic Interpretability](https://mlsanity.com/paper/2608.29034/) (2026)
- [Rate-Coding Bundle Memory: A Unified Model of Memory and Control for Symbolic Computation in the Brain](https://mlsanity.com/paper/2608.29189/) (2026)

## Abstract (arXiv)

Vector Symbolic Architectures (VSAs) are high-dimensional vector representations of objects (eg., words, image parts), relations (eg., sentence structures), and sequences for use with machine learning algorithms. They consist of a vector addition operator for representing a collection of unordered objects, a Binding operator for associating groups of objects, and a methodology for encoding complex structures. We first develop Constraints that machine learning imposes upon VSAs: for example, similar structures must be represented by similar vectors. The constraints suggest that current VSAs should represent phrases ("The smart Brazilian girl") by binding sums of terms, in addition to simply binding the terms directly. We show that matrix multiplication can be used as the binding operator for a VSA, and that matrix elements can be chosen at random. A consequence for living systems is that binding is mathematically possible without the need to specify, in advance, precise neuron-to-neuron connection properties for large numbers of synapses. A VSA that incorporates these ideas, MBAT (Matrix Binding of Additive Terms), is described that satisfies all Constraints. With respect to machine learning, for some types of problems appropriate VSA representations permit us to prove learnability, rather than relying on simulations. We also propose dividing machine (and neural) learning and representation into three Stages, with differing roles for learning in each stage. For neural modeling, we give "representational reasons" for nervous systems to have many recurrent connections, as well as for the importance of phrases in language processing. Sizing simulations and analyses suggest that VSAs in general, and MBAT in particular, are ready for real-world applications.