I don't think that octonions are the obvious next step here. They are a non-associative algebra, and therefore incredibly difficult to deal with.
Starting with real n-space, one can form the Clifford algebra, which essentially gives a method of multiplying vectors which "knows" something about the length and angle of vectors. The even subalgebra of the Clifford algebra gives a very convenient way of encoding rotations on real n-space. Furthermore, the Clifford algebra is always associative, and works for any n.
If you apply this construction for n=1, 2, 3, you get back the real numbers, complex numbers, and quaternions respectively. If you apply this for n=4, you get back an 8-dimensional associative algebra encoding rotations in 4-space.
Re:Deep Quaternion Networks (2017) [pdf]
by AstralStorm on Sunday March 18, 2018 @06:56AM
(#16611320)
Sounds quite quantum... negative time?
However, octonions are the obvious next step here: if you look at Appendix Figure 1, of "Deep Complex Networks" [1] , the authors authors used (Real + Complex), and Figure 1 of our paper[2] with quaternions uses (Real + Complex + Complex + Complex)!
[1] https://arxiv.org/pdf/1705.09792.pdf [2] https://arxiv.org/pdf/1712.04604.pdf