Collection of some quick ideas that I may or may not work on later.
1. Neural networks that take elements of a finite group as input
If we simply treat the input as general one-hot vectors the group structure will be lost. Could we somehow let the neural network be aware of the structure?
E.g. if the group is of order n then let the value of a neuron be a distribution on the n elements, and define a special type of neuron that takes two distributions P and Q and outputs R that is the distribution of c=ab^-1 where a~P and b~Q. We should be able to do this with some tensor arithmetic.
2. Iterative drawing for generating high-resolution images
What if instead of iteratively upsampling, we draw on a large canvas with progressively finer strokes?