Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

• URL: http://arxiv.org/abs/2603.11835v1
• Date: Thu, 12 Mar 2026 11:59:46 GMT
• Title: Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning
• Abstract summary: The most useful feature of quaternions lies in their ability to model three-dimensional rotations.<n>Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning.<n>This chapter lays down the foundation for this emerging research area.
• Score: 2.790230840980185
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the success of quaternions. The most useful feature of quaternions lies in their ability to model three-dimensional rotations which, in turn, have found various industrial applications such as in aeronautics and computergraphics. Recently, we have witnessed a renaissance of quaternions due to the rise of machine learning. To equip the reader to contribute to this emerging research area, this chapter lays down the foundation for: - augmented statistics for modelling quaternion-valued random processes, - widely linear models to exploit such advanced statistics, - quaternion calculus and algebra for algorithmic derivations, - mean square estimation for practical considerations. For ease of exposure, several examples are offered to facilitate the learning, understanding, and(hopefully) the adoption of this multidimensional domain.

Related papers

• A Family of Quaternion-Valued Differential Evolution Algorithms for Numerical Function Optimization [1.1470070927586018]
  We introduce a family of novel Quaternion-Valued Differential Evolution (QDE) algorithms that operate directly in the quaternion space.<n>Results show that our QDE variants achieve faster convergence and superior performance on several function classes.
  arXiv  Detail & Related papers  (2026-05-12T16:32:50Z)
• A Quantum of Learning: Using Quaternion Algebra to Model Learning on Quantum Devices [19.714874318004508]
  This article considers the problem of designing adaption and optimisation techniques for training quantum learning machines.<n>The division algebra of quaternions is used to derive an effective model for representing computation and measurement operations on qubits.
  arXiv  Detail & Related papers  (2025-04-17T14:51:21Z)
• Inferring Kernel $ε$-Machines: Discovering Structure in Complex Systems [49.1574468325115]
  We introduce causal diffusion components that encode the kernel causal-state estimates as a set of coordinates in a reduced dimension space. We show how each component extracts predictive features from data and demonstrate their application on four examples.
  arXiv  Detail & Related papers  (2024-10-01T21:14:06Z)
• Recent and Upcoming Developments in Randomized Numerical Linear Algebra for Machine Learning [49.0767291348921]
  Randomized Numerical Linear Algebra (RandNLA) is an area which uses randomness to develop improved algorithms for ubiquitous matrix problems. This article provides a self-contained overview of RandNLA, in light of these developments.
  arXiv  Detail & Related papers  (2024-06-17T02:30:55Z)
• The HR-Calculus: Enabling Information Processing with Quaternion Algebra [23.004932995116054]
  quaternions and their division algebra have proven to be advantageous in modelling rotation/orientation in three-dimensional spaces. adaptive information processing techniques specifically designed for quaternion-valued signals have only recently come to the attention of the machine learning, signal processing, and control communities.
  arXiv  Detail & Related papers  (2023-11-28T13:25:34Z)
• Experimental Results regarding multiple Machine Learning via Quaternions [1.2183405753834562]
  This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Based on quaternions and multiple machine learning algorithms, it has shown higher accuracy and significantly improved performance in prediction tasks.
  arXiv  Detail & Related papers  (2023-08-03T08:14:07Z)
• Higher-order topological kernels via quantum computation [68.8204255655161]
  Topological data analysis (TDA) has emerged as a powerful tool for extracting meaningful insights from complex data. We propose a quantum approach to defining Betti kernels, which is based on constructing Betti curves with increasing order.
  arXiv  Detail & Related papers  (2023-07-14T14:48:52Z)
• Quaternion-based machine learning on topological quantum systems [0.0]
  In this work, we incorporate quaternion algebras into data analysis to classify two-dimensional Chern insulators. For the unsupervised-learning aspect, we apply the principal component analysis (PCA) on the quaternion-transformed eigenstates. We construct our machine by adding one quaternion convolutional layer on top of a conventional convolutional neural network.
  arXiv  Detail & Related papers  (2022-09-29T05:02:20Z)
• Inducing Gaussian Process Networks [80.40892394020797]
  We propose inducing Gaussian process networks (IGN), a simple framework for simultaneously learning the feature space as well as the inducing points. The inducing points, in particular, are learned directly in the feature space, enabling a seamless representation of complex structured domains. We report on experimental results for real-world data sets showing that IGNs provide significant advances over state-of-the-art methods.
  arXiv  Detail & Related papers  (2022-04-21T05:27:09Z)
• Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels [108.60991563944351]
  We present a recipe for constructing gauge equivariant kernels, which induce vector-valued Gaussian processes coherent with geometry. We extend standard Gaussian process training methods, such as variational inference, to this setting.
  arXiv  Detail & Related papers  (2021-10-27T13:31:10Z)
• Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
  We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression. We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
  arXiv  Detail & Related papers  (2020-11-09T01:13:07Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.