# The Hyperdimensional Transform: a Holographic Representation of Functions

arXiv: https://arxiv.org/abs/2310.16065 · HTML: https://mlsanity.com/paper/2310.16065/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 46 references that are in the map.
- First posted on arXiv 24 Oct 2023.

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

- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [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)
- [A Neuro-vector-symbolic Architecture for Solving Raven's Progressive Matrices](https://mlsanity.com/paper/2203.04571/) (2022)
- [Neurocompositional computing: From the Central Paradox of Cognition to a new generation of AI systems](https://mlsanity.com/paper/2205.01128/) (2022)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [A theory of sequence indexing and working memory in recurrent neural networks](https://mlsanity.com/paper/1803.00412/) (2018)
- [An EMG Gesture Recognition System with Flexible High-Density Sensors and Brain-Inspired High-Dimensional Classifier](https://mlsanity.com/paper/1802.10237/) (2018)
- [Demeter: A Fast and Energy-Efficient Food Profiler using Hyperdimensional Computing in Memory](https://mlsanity.com/paper/2206.01932/) (2022)

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

- [The Hyperdimensional Transform for Distributional Modelling, Regression and Classification](https://mlsanity.com/paper/2311.08150/) (2023)
- [Hyperdimensional computing: a fast, robust and interpretable paradigm for biological data](https://mlsanity.com/paper/2402.17572/) (2024)

## Abstract (arXiv)

Integral transforms are invaluable mathematical tools to map functions into spaces where they are easier to characterize. We introduce the hyperdimensional transform as a new kind of integral transform. It converts square-integrable functions into noise-robust, holographic, high-dimensional representations called hyperdimensional vectors. The central idea is to approximate a function by a linear combination of random functions. We formally introduce a set of stochastic, orthogonal basis functions and define the hyperdimensional transform and its inverse. We discuss general transform-related properties such as its uniqueness, approximation properties of the inverse transform, and the representation of integrals and derivatives. The hyperdimensional transform offers a powerful, flexible framework that connects closely with other integral transforms, such as the Fourier, Laplace, and fuzzy transforms. Moreover, it provides theoretical foundations and new insights for the field of hyperdimensional computing, a computing paradigm that is rapidly gaining attention for efficient and explainable machine learning algorithms, with potential applications in statistical modelling and machine learning. In addition, we provide straightforward and easily understandable code, which can function as a tutorial and allows for the reproduction of the demonstrated examples, from computing the transform to solving differential equations.