I have recently come across the idea of encoding a 1D signal (i.e. a mono audio) as a complex vector instead of as a vector of reals, where the imaginary part is used to encode the cells' positions. It can be further generalized to a way of encoding a grayscale image into a vector of quaternions where imaginary parts now encode the pixels' positions. Here is the link to the paper discussing this way of encoding: https://arxiv.org/pdf/2006.08321.pdf The paper further goes on to octonions to encode data higher than 3D and concludes with geometric algebra.

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My question is on the implications of this kind of encoding for signal processing theory in general. Will this kind of encoding be useful in any application? Does it have any further implications?

  • $\begingroup$ "what are the implications of [mathematical concept]" is simply far too broad and/or asking for opinions. I don't see how there's a "correct" answer to your question... The "interpretation" as complex value instead of a (value, position) tuple doesn't contain any advantage; neither to the geometry of the problem, nor to neural networks, as very very vaguely handwaved at in the paper... So, I'm not usually this negative, but this yields no advantage that I can think of. $\endgroup$ – Marcus Müller Jun 16 at 15:37
  • $\begingroup$ In fact, the authors even state that you need to convert the complex or hypercomplex thing back to a real projection so to work with it; and that's exactly where you've started from. So, really, nothing but a linear operation on input vectors. And guess what, that's what NNs have always been doing. $\endgroup$ – Marcus Müller Jun 16 at 15:39
  • $\begingroup$ But would not the real projection (if done correctly) carry the extra information of cell positions in some form so is more informative than the original vector which has no information on cell positions whatsoever? $\endgroup$ – user Jun 16 at 15:47
  • $\begingroup$ "the original vector has no information on positions" that's simply not true. It does. These values aren't unordered, they are ordered. The position is implicit. And that's why we use CNNs, anyway... $\endgroup$ – Marcus Müller Jun 16 at 15:51
  • $\begingroup$ The authors also state that you can directly work with complex valued neural networks, I am not sure whether any experimentation done in this regard. I guess what I am trying to understand is considering a signal as a complex/hypercomplex vector be more appropriate than just seeing it as a vector (as referred in linear algebra) because a vector does not have any cell positions encoded. $\endgroup$ – user Jun 16 at 15:56