I have been experimenting with sub-random patterns to combine with the fast (Walsh) Hadamard transform to create sub-random projections. With a pure random projection the response to a local pattern is completely different in every location that pattern can be in. That is a bit extreme for some applications. I picked a very simple system based on the sign of the sine of a linear sequence. It is not as sensitive to its parameters as a factal and not as chaotic but nearly:
It might be worthwhile constructing an associative memory out of such sub-random projections and then using that to learn classification problems. That is an experiment to do. The idea being you might get some translation and perhaps scaling and rotation invariance.