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Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

Sayed Pouria Talebi, Clive Cheong Took

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.

Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

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.
Paper Structure (15 sections, 140 equations, 3 figures, 4 tables)

This paper contains 15 sections, 140 equations, 3 figures, 4 tables.

Table of Contents

  1. Hypercomplex Widely Linear Processing
  2. Introduction
  3. Quaternion Algebra
  4. Three-Dimensional Rotations
  5. Quaternion Involutions
  6. The Augmented Quaternion Approach
  7. Augmented Statistics
  8. The Autocorrelation Matrices
  9. Duality between auto-correlation matrices in $\mathbb{R}$ and $\mathbb{H}$
  10. MMSE Estimation and Widely linear Model
  11. Quaternion Calculus
  12. Multiplication Rules
  13. The Product Rule
  14. Chain Derivative Rule in the $\mathbb{HR}$-Calculus
  15. Summary

Figures (3)

  • Figure 1: Schematic of a rotation around $\eta$ by an angle of $\theta$, with $q_{\text{pre}}$ and $q_{\text{post}}$ pointing to the pre- and post-rotation orientation of the object in question.
  • Figure 2: Schematic demonstrating $q\in\mathbb{H}$ along side its involution $q^{\kappa}$. The real axis has been omitted for simplicity as $\Re\{q\}$ does not change under quaternion involutions. In the top left graph, $q$ (in black), its projection onto the $\imath-\jmath$ plane (in blue), and its projection onto the $\kappa$ axis (in red) is shown. In the top right graph, projection of $q$ onto the $\imath-\jmath$ plane is rotated around the centre by $\pi$. In the bottom left graph, $q^{\kappa}$ is constructed from the projection of $q$ onto the $\kappa$ axis and the rotated projection of $q$ onto the $\imath-\jmath$ plane. In the bottom right graph, all elements of $q$ and $q^{\kappa}$ are shown together.
  • Figure 3: Absolute magnitude of autocorrelations as a function of lag. Bottom row: The left plot shows the non-symmetric pseudo-autocorrelation of full quaternion sequence, whereas the right plot shows its symmetric pseudo-autocorrelation, if its imaginary part was considered as pure quaternions.