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.

超维变换是一种新型的积分变换，将方块可积函数转换为噪声鲁棒、全息、高维表示的超维向量。它提供了辅助其他积分变换的理论基础和见解，与傅里叶、拉普拉斯和模糊变换密切相关。此外，它为超维计算领域提供了强大而灵活的框架，该计算范式在高效和可解释的机器学习算法方面引起了广泛关注，具有在统计建模和机器学习中应用的潜力。

超维变换：函数的全息表示