Hi

[PallHaraldsson]

I wanted to tell you I pinged you at:

https://discourse.julialang.org/t/quaternion-and-up-to-sedenion-valued-neural-networks-parallellizing-hamilton-product-on-gpus-cuda/48442/2?u=palli

in the thread I started (I probably have the answers I need for now, so I'm only telling you I had you and still have you in mind).

This package may not be the best way to contact you, but I noticed on reddit you being banned on discourse (I believe it was from your post), so I wasn't sure you would actually see the ping.

At least you couldn't answer (there, or maybe not knowing you/if to contact me).

I recall, in some thread I started, you saying stuff related to neural networks.

I'm up-to-speed on complex number, and enough up to sedenions (and have ideas how to make faster), while there's probably stuff I need to know later.

Does this have a lot to do with your work, e.g. this package?

I'm currently looking into these papers:

Quaternion Fourier Transform on Quaternion Fields and Generalizations

A Generalization of the Octonion Fourier Transform to 3-D Octonion-Valued Signals -- Properties and Possible Applications to 3-D LTI Partial Differential Systems

I knew of some generalizations of Fourier transform, i.e. to non-uniform or hexagonal, but not to hyper-complex before.

I googled "sedenion fourier transform" to see if also existing found nothing except for "Hey there! Quaternion and Clifford Fourier Transforms and Wavelets searched all the web couldn't find anywhere."

I did not find anything on trigintaduonions fourier transform either, but I did find intriguing paper from this year:

https://www.researchgate.net/publication/341626625_Theory_of_Trigintaduonion_Emanation_and_Origins_of_alpha_and_pi

The new unification approach described here gives a precise derivation for the mysterious physics constant alpha (the fine-structure constant) from the mathematical physics formalism providing maximal information propagation, with alpha being the maximal perturbation amount. Furthermore, the new unification provides that the structure of the space of initial 'propagation' (with initial propagation being referred to as 'emanation') has a precise derivation, with a unit-norm perturbative limit that leads to an iterative-map-like computed alpha (a limit that is precisely related to the Feigenbaum bifurcation constant and thus fractal). The computed alpha can also, by a maximal information propagation argument, provide a derivation for the mathematical constant pi.

[chakravala]

Thanks for opening the issue, I cannot respond on the discourse right now and do not receive any pings from there either, i only saw your post because i checked manually.

Regarding Quaternion Neural networks, I have been at some conference sessions where this was the main topic. I also know of some people doing it with python and tensor flow, they are not using naive implementations, just FYI. If this interests you, I can try to find the relevant references, but you'll need to wait until next week for that, i dont have much computer time the next few days.

As for the topics you raise, Octonions are more weird, they can be found in a 7-dimensional geometric algebra or as a nested algebra of quaternions, but I havent taken the time really bother with Octonions for my own purposes. I explored it a bit.. but abandoned it since there are a lot of other important things to work on also, yet i'd be interested in taking another look at this again.

As for the fine structure constant, thats very interesting. Another person I know Christoph Schiller is working on that from a different view https://www.motionmountain.net

I would like to take a look at all those references and keep up with your findings, so please feel free to post more comments here, but the next days I am not available.

[PallHaraldsson]

Quaternion networks still interest me, I mean since two days ago when I first heard of them. :)

I'm not sure you read my thread. but if numbers of parameters roughly half, going to complex, and again and again up to sedenions(?), then I'm most interested in sedenions, or what comes after, and you can go infinity high. This sedenion-valued neural network from August 2020 is the first NN based on them, and I didn't really know of any applications for them NN or not, let alone higher order.

I'm just not sure about the downside of going higher to say octonions, and if it hinders some applications of neural networks. The universal approximation theory still applies, going up, I guess infinity, but there might be wasted dimensions/calculations?

Yes, the paper on alpha seems very interesting, I thought it was unexplained still, probably no other have, and that also implies this math has some (more value) I'm just not sure what it means about reality, the paper mentioned string theory "emergent", so the real world may not need that many dimension? The fundamental theory, or maybe going the other direction, is it infinite-dimensional?

You may want to look into Geometric Unity, since you seems interested in physics, as I thought you would. I did watch the whole Oxford lecture on it (and some of the interviews with him), while understanding almost nothing, to no real depth (and the same for your video), but I have a feeling you would and the math is similar or at least overlapping. That guy also reminds me of you, an outsider.