# Computing with Residue Numbers in High-Dimensional Representation

arXiv: https://arxiv.org/abs/2311.04872 · HTML: https://mlsanity.com/paper/2311.04872/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 73 references that are in the map.
- First posted on arXiv 8 Nov 2023.

## Builds on (most cited first, 17 of 17)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges](https://mlsanity.com/paper/2112.15424/) (2021)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [Vector Symbolic Architectures as a Computing Framework for Emerging Hardware](https://mlsanity.com/paper/2106.05268/) (2021)
- [A theory of sequence indexing and working memory in recurrent neural networks](https://mlsanity.com/paper/1803.00412/) (2018)
- [Robust computation with rhythmic spike patterns](https://mlsanity.com/paper/1901.07718/) (2019)
- [Understanding Hyperdimensional Computing for Parallel Single-Pass Learning](https://mlsanity.com/paper/2202.04805/) (2022)
- [Neuromorphic Visual Scene Understanding with Resonator Networks](https://mlsanity.com/paper/2208.12880/) (2022)
- [Density Encoding Enables Resource-Efficient Randomly Connected Neural Networks](https://mlsanity.com/paper/1909.09153/) (2019)
- [Generalized Learning Vector Quantization for Classification in Randomized Neural Networks and Hyperdimensional Computing](https://mlsanity.com/paper/2106.09821/) (2021)
- [In-memory factorization of holographic perceptual representations](https://mlsanity.com/paper/2211.05052/) (2022)
- [Integer Factorization with Compositional Distributed Representations](https://mlsanity.com/paper/2203.00920/) (2022)
- [Probable nature of higher-dimensional symmetries underlying mammalian grid-cell activity patterns](https://mlsanity.com/paper/1411.2136/) (2014)
- [Efficient Decoding of Compositional Structure in Holistic Representations](https://mlsanity.com/paper/2305.16873/) (2023)

## Built on by (most cited first, 7 of 7)

- [Compositional Factorization of Visual Scenes with Convolutional Sparse Coding and Resonator Networks](https://mlsanity.com/paper/2404.19126/) (2024)
- [A Circular Argument : Does RoPE need to be Equivariant for Vision?](https://mlsanity.com/paper/2511.08368/) (2025)
- [Hyperdimensional computing: a fast, robust and interpretable paradigm for biological data](https://mlsanity.com/paper/2402.17572/) (2024)
- [Hey Pentti, We Did It Again!: Differentiable vector-symbolic types that prove polynomial termination](https://mlsanity.com/paper/2510.16533/) (2025)
- [Do traveling waves make good positional encodings?](https://mlsanity.com/paper/2511.11668/) (2025)
- [Robot Arm Control via Cognitive Map Learners](https://mlsanity.com/paper/2603.26773/) (2026)
- [Real-Valued Hyperdimensional Sequence Representations with Hadamard Product Binding and Shift Equivariance](https://mlsanity.com/paper/2608.28334/) (2026)

## Abstract (arXiv)

We introduce Residue Hyperdimensional Computing, a computing framework that unifies residue number systems with an algebra defined over random, high-dimensional vectors. We show how residue numbers can be represented as high-dimensional vectors in a manner that allows algebraic operations to be performed with component-wise, parallelizable operations on the vector elements. The resulting framework, when combined with an efficient method for factorizing high-dimensional vectors, can represent and operate on numerical values over a large dynamic range using vastly fewer resources than previous methods, and it exhibits impressive robustness to noise. We demonstrate the potential for this framework to solve computationally difficult problems in visual perception and combinatorial optimization, showing improvement over baseline methods. More broadly, the framework provides a possible account for the computational operations of grid cells in the brain, and it suggests new machine learning architectures for representing and manipulating numerical data.