Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

Sayed Pouria Talebi; Clive Cheong Took

stat.MLMar 12, 2026

Hypercomplex Widely Linear Processing: Fundamentals for Quaternion Machine Learning

Authors: Sayed Pouria Talebi, Clive Cheong Took

Organizations: Computer Science Department, University of Roehampton, London WC2R-2LS, U.K. · Electronic Engineering Department, Royal Holloway University of London, TW20 0EX, U.K.

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.

Figures & tables

$\mathbb{N}$	set of natural numbers
$\mathbb{R}$	set of real-valued numbers
$\mathbb{R}^{+}$	set of positive real-valued numbers
$\mathbb{C}$	set of complex-valued numbers
$\mathbb{H}$	set of quaternion-valued numbers
$\Re\{\cdot\}$	operator returning the real component
$\Im\{\cdot\}$	operator returning the imaginary component

Table 1

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.

Autocorrelation matrices in $\mathbb{H}$ (of size $3\times 3$ )
$\mathbf{R}_{c}(n)=\left[\begin{array}[]{ccc}{\color[rgb]{0.25,0.25,0.25}73}&15.33-13.33\imath&{\color[rgb]{0.5,0.5,0.5}18.67-10.33\imath}\\ &+0.33\jmath+9\kappa&{\color[rgb]{0.5,0.5,0.5}+5.33\jmath-10\kappa}\\ 15.33+13.33\imath&{\color[rgb]{0.5,0.5,0.5}73}&15.33-13.33\imath\\ -0.33\jmath-9\kappa&&+0.33\jmath+9\kappa\\ {\color[rgb]{0.5,0.5,0.5}18.67+10.33\imath}&15.33+13.33\imath&{\color[rgb]{0.5,0.5,0.5}73}\\ {\color[rgb]{0.5,0.5,0.5}-5.33\jmath+10\kappa}&-0.33\jmath-9\kappa&\end{array}\right]$
$\mathbf{R}_{\imath}(n)=\left[\begin{array}[]{ccc}{\color[rgb]{0.5,0.5,0.5}35+4\jmath-6.67\kappa}&31.33-1.33\imath&{\color[rgb]{0.5,0.5,0.5}17.33-13\imath}\\ &+22.33\jmath+19.67\kappa&{\color[rgb]{0.5,0.5,0.5}-0.67\jmath-10.67\kappa}\\ 31.33+1.33\imath&{\color[rgb]{0.5,0.5,0.5}35+4\jmath-6.67\kappa}&31.33-1.33\imath\\ +22.33\jmath-19.67\kappa&&+22.33\jmath+19.67\kappa\\ {\color[rgb]{0.5,0.5,0.5}17.33+13\imath}&31.33+1.33\imath&{\color[rgb]{0.5,0.5,0.5}35+4\jmath-6.67\kappa}\\ {\color[rgb]{0.5,0.5,0.5}-0.67\jmath-10.67\kappa}&+22.33\jmath+19.67\kappa&\end{array}\right]$
$\mathbf{R}_{\jmath}(n)=\left[\begin{array}[]{ccc}{\color[rgb]{0.5,0.5,0.5}-28.33+14.67\imath-5.33\kappa}&-24.67+20.67\imath&{\color[rgb]{0.5,0.5,0.5}-14+14.33\imath}\\ &-15\jmath-29.67\kappa&{\color[rgb]{0.5,0.5,0.5}-4.67\jmath+12.67\kappa}\\ -24.67+20.67\imath&{\color[rgb]{0.5,0.5,0.5}-28.33+14.67\imath-5.33\kappa}&-24.67+20.67\imath\\ +15\jmath-29.67\kappa&&-15\jmath-29.67\kappa\\ {\color[rgb]{0.5,0.5,0.5}-14+14.33\imath}&-24.67+20.67\imath&{\color[rgb]{0.5,0.5,0.5}-28.33+14.67\imath-5.33\kappa}\\ {\color[rgb]{0.5,0.5,0.5}+4.67\jmath+12.67\kappa}&+15\jmath-29.67\kappa&\end{array}\right]$
$\mathbf{R}_{\kappa}(n)=\left[\begin{array}[]{ccc}{\color[rgb]{0.5,0.5,0.5}-51.67+36\imath-5.33\jmath}&-8.67+12.67\imath&{\color[rgb]{0.5,0.5,0.5}-16.67+15.67\imath}\\ &-7.67\jmath-5.67\kappa&{\color[rgb]{0.5,0.5,0.5}-4\jmath+6.67\kappa}\\ -8.67+12.67\imath&{\color[rgb]{0.5,0.5,0.5}-51.67+36\imath-5.33\jmath}&-8.67+12.67\imath\\ -7.67\jmath+5.67\kappa&&-7.67\jmath-5.67\kappa\\ {\color[rgb]{0.5,0.5,0.5}-16.67+15.67\imath}&-8.67+12.67\imath&{\color[rgb]{0.5,0.5,0.5}-51.67+36\imath-5.33\jmath}\\ {\color[rgb]{0.5,0.5,0.5}-4\jmath-6.67\kappa}&-7.67\jmath+5.67\kappa&\end{array}\right]$

Table 2: Quaternion-valued autocorrelation matrices considered in Example 3.2 . Observe the Hermitian structure of

{R}_{c}(n)

as shown in ( 49 ) and the

\eta-

Hermitian property of

\mathbf{R}_{\imath}(n)

\mathbf{R}_{\jmath}(n)

, and

\mathbf{R}_{\kappa}(n)

as illustrated in ( 46 ).

Auto-correlation	$\mathbf{R}_{c}(n)$	$\mathbf{R}_{\imath}(n)$
matrix
$\Re\{\cdot\}$	$\mathbf{R}_{rr}(n)+\mathbf{R}_{\imath\imath}(n)+\mathbf{R}_{\jmath\jmath}(n)+\mathbf{R}_{\kappa\kappa}(n)$	$\mathbf{R}_{rr}(n)+\mathbf{R}_{\imath\imath}(n)-\mathbf{R}_{\jmath\jmath}(n)-\mathbf{R}_{\kappa\kappa}(n)$
$\Im_{\imath}\{\cdot\}$	$\mathbf{R}_{\imath r}(n)-\mathbf{R}_{r\imath}(n)+\mathbf{R}_{\kappa\jmath}(n)-\mathbf{R}_{\jmath\kappa}(n)$	$\mathbf{R}_{\imath r}(n)-\mathbf{R}_{r\imath}(n)+\mathbf{R}_{\jmath\kappa}(n)-\mathbf{R}_{\kappa\jmath}(n)$
$\Im_{\jmath}\{\cdot\}$	$\mathbf{R}_{\jmath r}(n)-\mathbf{R}_{r\jmath}(n)+\mathbf{R}_{\imath\kappa}(n)-\mathbf{R}_{\kappa\imath}(n)$	$\mathbf{R}_{r\jmath}(n)+\mathbf{R}_{\jmath r}(n)-\mathbf{R}_{\kappa\imath}(n)-\mathbf{R}_{\imath\kappa}(n)$
$\Im_{\kappa}\{\cdot\}$	$\mathbf{R}_{\kappa r}(n)-\mathbf{R}_{r\kappa}(n)+\mathbf{R}_{\jmath\imath}(n)-\mathbf{R}_{\imath\jmath}(n)$	$\mathbf{R}_{\kappa r}(n)+\mathbf{R}_{r\kappa}(n)+\mathbf{R}_{\imath\jmath}(n)+\mathbf{R}_{\jmath\imath}(n)$

Table 3: Quaternion-valued auto-correlation matrices in terms of their real-valued counterparts

Auto-correlation	$\mathbf{R}_{\jmath}(n)$	$\mathbf{R}_{\kappa}(n)$
matrix
$\Re\{\cdot\}$	$\mathbf{R}_{rr}(n)-\mathbf{R}_{\imath\imath}(n)+\mathbf{R}_{\jmath\jmath}(n)-\mathbf{R}_{\kappa\kappa}(n)$	$\mathbf{R}_{rr}(n)-\mathbf{R}_{\imath\imath}(n)-\mathbf{R}_{\jmath\jmath}(n)+\mathbf{R}_{\kappa\kappa}(n)$
$\Im_{\imath}\{\cdot\}$	$\mathbf{R}_{\imath r}(n)+\mathbf{R}_{r\imath}(n)+\mathbf{R}_{\kappa\jmath}(n)+\mathbf{R}_{\jmath\kappa}(n)$	$\mathbf{R}_{\imath r}(n)+\mathbf{R}_{r\imath}(n)-\mathbf{R}_{\jmath\kappa}(n)-\mathbf{R}_{\kappa\jmath}(n)$
$\Im_{\jmath}\{\cdot\}$	$\mathbf{R}_{\jmath r}(n)-\mathbf{R}_{r\jmath}(n)+\mathbf{R}_{\kappa\imath}(n)-\mathbf{R}_{\imath\kappa}(n)$	$\mathbf{R}_{r\jmath}(n)+\mathbf{R}_{\jmath r}(n)+\mathbf{R}_{\kappa\imath}(n)+\mathbf{R}_{\imath\kappa}(n)$
$\Im_{\kappa}\{\cdot\}$	$\mathbf{R}_{\kappa r}(n)+\mathbf{R}_{r\kappa}(n)-\mathbf{R}_{\imath\jmath}(n)-\mathbf{R}_{\jmath\imath(n)}$	$\mathbf{R}_{\kappa r}(n)-\mathbf{R}_{r\kappa}(n)+\mathbf{R}_{\imath\jmath}(n)-\mathbf{R}_{\jmath\imath}(n)$

Table 4: Quaternion-valued auto-correlation matrices in terms of their real-valued counterparts

Autocorrelation matrices in $\mathbb{R}$ (of size $3\times 3$ )
$\mathbf{R}_{rr}(n)=\left[\begin{array}[]{ccc}7&3.33&1.33\\ 3.33&7&3.33\\ 1.33&3.33&7\end{array}\right]$	$\mathbf{R}_{\imath\imath}(n)=\left[\begin{array}[]{ccc}47&20&16.67\\ 20&47&20\\ 16.67&20&47\end{array}\right]$
$\mathbf{R}_{\jmath\jmath}(n)=\left[\begin{array}[]{ccc}15.33&-8&1\\ -8&15.33&-8\\ 1&-8&15.33\end{array}\right]$	$\mathbf{R}_{\kappa\kappa}(n)=\left[\begin{array}[]{ccc}3.67&0&-0.33\\ 0&3.67&0\\ -0.33&0&3.67\end{array}\right]$
$\mathbf{R}_{\imath r}(n)=\left[\begin{array}[]{ccc}12.67&4.67&1.67\\ 12&12.67&4.67\\ 13.33&12&12.67\end{array}\right]$	$\mathbf{R}_{\jmath r}(n)=\left[\begin{array}[]{ccc}-0.33&0&-1\\ 7.33&-0.33&0\\ -1.33&7.33&-0.33\end{array}\right]$
$\mathbf{R}_{\kappa r}(n)=\left[\begin{array}[]{ccc}-3&-1.67&-0.33\\ -3.33&-3&-1.67\\ 1.33&-3.33&-3\end{array}\right]$	$\mathbf{R}_{\jmath\imath}(n)=\left[\begin{array}[]{ccc}-0.33&16&-10\\ 8.67&-0.33&16\\ -1.67&8.67&-0.33\end{array}\right]$
$\mathbf{R}_{\kappa\imath}(n)=\left[\begin{array}[]{ccc}-2.33&-11.33&-3.33\\ -3.67&-2.33&-11.33\\ 1.67&-3.67&-2.33\end{array}\right]$	$\mathbf{R}_{\kappa\jmath}(n)=\left[\begin{array}[]{ccc}-5.33&-1&0.33\\ 5&-5.33&-1\\ -1&5&-5.33\end{array}\right]$

Table 5: Real-valued autocorrelation matrices corresponding to their quaternion-valued counterparts in Table 2 . The double subscripts indicate the correlation between the two relevant quaternion components. For example,

R_{\imath r}(n)

corresponds to the cross correlation between the

\imath-

imaginary and real component.

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