If we have a variable $X \in R^n$ (like an image) and a non-linear, invertible function $F: X \rightarrow H$ that maps $R^n$ to $R^d$ with $(d<n)$, how can I learn a F that preserves measure?
Particularly, if $X$ follows a probability distribution $p(X)$, can I analytically compute $p(X)$ from $p(H)$ using change-of-variables formula?
In my problem, F is a normalizing flow composed by a series of $k$ invertible transformations $f_k: R^n \rightarrow R^n$, and a function $proj(H)$ that takes only d components of it. Next, I have another normalizing flow $G: proj(H) \rightarrow Z$, with $p(Z)$ defined as $\mathbf{N}(0,I)$, and trained by minimizing negative log-likelihood.
How can I restrict F and G to preserve measure?