Sensitivity analysis - access the jacobian

[MolinAlexei]

I am trying to evaluate the relevance of a summary statistic I am using. My logic is the following :
Given the model producing the observable $m(\theta, \alpha) \rightarrow X$, with $\theta$ the parameters I am doing inference on, $X$ the observable and $\alpha$ an internal nuisance parameter of the model (e.g. measurement error),
If this model is a bijection from the space of possible $\theta$, let's call it $\Omega$, to the image, let's call it $m(\Omega)$ for a fixed parameter $\alpha$, then the observable I am using is ideal in the sense that running inference with sbi should give me the smallest possible posterior distribution of $\theta$, given the distribution of $\alpha$. (And on the contrary, if $m$ is not a bijection, that means that my observable will give me degeneracies in the input parameters, or that I will not be able to put the best constraint on some of them)
Then, I would want to evaluate the bijectivity of m. It should be automatically surjective from $\Omega$ to $m(\Omega)$. The injectivity can be evaluated by the jacobian - which is already computed as part of the Hessian in the tutorial "Active subspaces for sensitivity analysis". If I have access to the jacobian and if I can show that its rank is $N$ with $N$ the dimension of $\theta$, then I should have proven that $m$ is injective.

This should give information as to whether the summary statistic that I chose as my observable is a good way to retrieve the input parameters, or if it has inherent degeneracies that prevent me from retrieving the best possible posterior on the parameters.

I hope I am making sense, and if so, I hope it can be implemented in a similar fashion to the sensitivity analysis.

Edit : after doing some research I realize now that the information I am looking for is best described by the Fisher matrix, which is already evaluated in the sensitivity analysis as $\mathbb{E}_{p(\theta|x_0)} [ \nabla p(\theta|x_0)^T \nabla p(\theta | x_0)]$. Then, by evaluating this term for posterior log probability obtained with and without the summary statistics I can compare how close my summary statistic is ideal by comparing the determinants of the Fisher matrices.