I am currently reading up on RealNVP, which has the following transformations according Lilian Weng:
$$ \begin{aligned} \mathbf{y}_{1:d} &= \mathbf{x}_{1:d} \\ \mathbf{y}_{d+1:D} &= \mathbf{x}_{d+1:D} \odot \exp({s(\mathbf{x}_{1:d})}) + t(\mathbf{x}_{1:d}) \end{aligned} $$
I am wondering how the inverse has the scale and shift networks simply reparameterized by the outputs $\mathbf{y}_i$'s instead. Wouldn't the learned weights only provide mappings from $\mathbf{x} \rightarrow \mathbf{y}$?
Here is the math to show invertibility: $$ \begin{cases} \mathbf{y}_{1:d} &= \mathbf{x}_{1:d} \\ \mathbf{y}_{d+1:D} &= \mathbf{x}_{d+1:D} \odot \exp({s(\mathbf{x}_{1:d})}) + t(\mathbf{x}_{1:d}) \end{cases} \Leftrightarrow \begin{cases} \mathbf{x}_{1:d} &= \mathbf{y}_{1:d} \\ \mathbf{x}_{d+1:D} &= (\mathbf{y}_{d+1:D} - t(\mathbf{y}_{1:d})) \odot \exp(-s(\mathbf{y}_{1:d})) \end{cases} $$