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Since both sign flipping and the WHT leave vector length unchanged
a non-zero vector repeatedly passed through such a random projection
oscillates with a constant vector amplitude.
An identical random projection can act as a resonator in such a system.
Again feeding the output back to the input.
Passing data from just one dimension of the oscillator to the same
dimension in a resonator forces resonance build up of vector amplitude.
Observe the y-axis scale.
You can create a damped oscillator/resonator by
scalar multiplying the vector by a value less than one each trip around.
It can be a good idea to use random permutations in addition to sign flipping.
Such resonators can act as a reservoir in reservoir computing.
There is a good question about feeding sparse data into such a resonator.
Does that lead to a concentration of information in the system?
And what are the limits of that?
The system also acts as set of "digital signal processing" filters
that could offer some of the same benefits as convolution in deep neural
networks.
Finally connecting such resonators together across one or few dimensions
produces effects very similar to entropy in physical systems.
Like connecting containers of gasses at different pressures together.