@article{01JNNB1FBVCQD8W0MNTD2D61GF,
  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.}},
  author       = {{Dewulf, Pieter and Stock, Michiel and De Baets, Bernard}},
  issn         = {{1932-4553}},
  journal      = {{IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING}},
  keywords     = {{Transforms,Vectors,Kernel,Stochastic processes,Machine learning,Encoding,Laplace equations,Integral transforms,differential equations,hyperdimensional computing,machine learning,efficient computing}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{3--18}},
  title        = {{The hyperdimensional transform : a holographic representation of functions}},
  url          = {{http://doi.org/10.1109/jstsp.2024.3405850}},
  volume       = {{19}},
  year         = {{2025}},
}

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