Gan paper: sampling the distribution

Question

In the Gan paper it is said page 3 Figure 1:

"The lower horizontal line is the domain from which z is sampled, in this case uniformly. The horizontal line above is part of the domain of x. The upward arrows show how the mapping x = G(z) imposes the non-uniform distribution pg on transformed samples"

For those who wants to see the figure:

I wanted to know what that would mean in practical case. Let's say you are working with images that are normalized between [0,..,1] this would be the domain of x as referred in the paper right? Does this mean that I would have to sample my z from the domain of x, i.e: [0,..,1] ?

In most implementations I see people taking point randomly using things such as: np.random.randn(latent_dim)

Answer 1

Let's say you are working with images that are normalized between [0,..,1] this would be the domain of x as referred in the paper right?

No, the domain of X would be "images of [whatever they contain (e.g. dogs)] normalized between 0 and 1".

Does this mean that I would have to sample my z from the domain of x, i.e: [0,..,1] ?

No, they are both different domains and the generator $G$ maps between them.

In the paragraph you linked, the authors just point out that the generator $G$ is a function that maps the input data (i.e. random vectors following a uniform distribution in $[0, 1]$) to the output data (e.g. images of dogs) and that the mapping is non-regular, meaning that very different inputs may lead to similar outputs and vice versa.