Why aren't Normalizing Flows suitable for Discrete Distributions? I am currently trying to understand why normalizing flows are not applicable to discrete distributions (a quick primer on NF can be found here).
The assumptions on the transformation f  between the probability distributions are:

*

*f must be an invertible function

*f must be a smooth function

Assume I want to learn a normalizing flow between a Poisson and a Normal distribution.
If I discretize my Normal distribution, both have an infinite support, i.e., the same number of support elements and hence, I can find an invertible mapping between them (another option would be to consider only the first n natural numbers for the Poisson distribution and also select n elements from the Normal distribution).
Moreover, when I am training a Normalizing Flow on my computer, I never have continuous data - instead, I have discrete samples from my distribution. So, where is the issue with taking these samples from a discrete distribution now? Unfortunately, I cannot see the difference to the case where I am trying to map between two Normal distributions: if I sample from a Normal distribution I have discrete samples as well.
Thanks in advance!

An NF is essentially just a change of variable. If you want to change a density supported on a discrete set, to a density supported on a continuous set, the corresponding transformation is bound to degenerate (which manifests as a Jacobian determinant of infinity or 0, depending which direction of the flow you consider). Actually, even training a NF for data living on a low-dimensional manifold will lead to the same problem, thus this degeneracy problem is not only limited to discrete-continuous case, see e.g. https://proceedings.neurips.cc/paper/2021/file/4c07fe24771249c343e70c32289c1192-Paper.pdf
