It turns out a lot of disconnected spaces can be approximated by smooth ones that have really sharp boundaries, which more or less seems to be how neural networks will approximate something like discrete tokens
Can be approximated yes. Approximated well? No, but you can get away with it sometimes with saturation functions like softmax. But badly. It’s like trying to solve an integer program as a linear program. You end up with a relaxation that is not integral and not the real answer.
An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.
An integer lattice can only be a manifold in a trivial sense (dimension 0). But not for any positive dimensions.