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by wespiser_2018
3008 days ago
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Quaternions are known to represent spatial transforms, and there is a little bit of prior work that demonstrates quaternion filters 'make sense'. 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 |
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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.