Assume in a normalizing flow model we have the following forward transformation given.
$$
f(z)=
\begin{bmatrix}
z_1 +2z_2\\
3z_1 + 4z_2\\
z_3 – 2z_4 \\
3z_3 -4z_4
\end{bmatrix}
$$
If a transformation is not invertible, it's easy to show, i.e., we show that the function is not bijective. Now, for this one, I am sure it's invertible and my way of "showing" it would be solving the linear system of equations to get the inverse.
My questions are:
- Is there a sufficient condition to show a higher-dimensional function is invertible?
- Is it sufficient to calculate the inverse here to prove $f$ has one? I have made mistakes before where I "calculated the inverse" for non-invertible functions. Technically, it worked out, but of course it was wrong.
- Are there tools available which quickly do this for me?
Best Answer
You appear to be looking for the implicit function theorem (which actually also works perfectly well for functions that have an explicit inverse). This theorem gives you sufficient conditions for the invertability of your multivariate function. For a multivariate function with the same number of inputs and outputs, the theorem says that if the Jabocian determinant is non-zero then the function is invertible. In the present case you have:
$$\mathbf{J}(\mathbf{z}) = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 3 & 4 & 0 & 0 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 3 & -4 \\ \end{bmatrix} \quad \quad \quad \det \mathbf{J}(\mathbf{z}) = -4 \neq 0,$$
so the function is invertible.