The Hyperdimensional Transform: a Holographic Representation of Functions

TL;DR

The paper introduces the hyperdimensional transform, a novel integral transform that maps functions into noise-robust, high-dimensional representations, connecting with classical transforms and advancing hyperdimensional computing for machine learning.

Contribution

It formally defines the hyperdimensional transform, explores its properties, and demonstrates its applications in statistical modeling and differential equations, providing theoretical and practical insights.

Findings

Hyperdimensional transform produces noise-robust, holographic representations.

It connects with Fourier, Laplace, and fuzzy transforms.

Code implementation enables practical application and reproduction.

Abstract

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.