Hyperdimensional computing (HDC) is an increasingly popular computing
paradigm with immense potential for future intelligent applications. Although
the main ideas already took form in the 1990s, HDC recently gained significant
attention, especially in the field of machine learning and data science. Next
to efficiency, interoperability and explainability, HDC offers attractive
properties for generalization as it can be seen as an attempt to combine
connectionist ideas from neural networks with symbolic aspects. In recent work,
we introduced the hyperdimensional transform, revealing deep theoretical
foundations for representing functions and distributions as high-dimensional
holographic vectors. Here, we present the power of the hyperdimensional
transform to a broad data science audience. We use the hyperdimensional
transform as a theoretical basis and provide insight into state-of-the-art HDC
approaches for machine learning. We show how existing algorithms can be
modified and how this transform can lead to a novel, well-founded toolbox. Next
to the standard regression and classification tasks of machine learning, our
discussion includes various aspects of statistical modelling, such as
representation, learning and deconvolving distributions, sampling, Bayesian
inference, and uncertainty estimation.

超维计算 (HDC) 是一种具有巨大潜力的计算范例，尤其在机器学习和数据科学领域引起了广泛关注。本文介绍了超维变换的理论基础，并展示了其在机器学习中的应用，包括回归、分类、统计建模等方面。

超维变换在分布建模、回归和分类中的应用

The Hyperdimensional Transform for Distributional Modelling, Regression  and Classification

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。