Is unsupervised disentanglement really impossible? In Locatello et al's Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations he claims to prove unsupervised disentanglement is impossible.  

His entire claim is founded on a theorem (proven in the appendix) that states in my own words:    

Theorem: for any distribution $p(z)$ where each variable $z_i$ are  independent of each other there exists an infinite number of transformations $\hat z = f(z)$ from $\Omega_z \rightarrow \Omega_z$ with distribution $q(\hat z$) such that all variables $\hat z_i$ are entangled/correlated and the distributions are equal ($q(\hat z) = p(z)$)  

Here is the exact wording from the paper:

  

(I provide both because my misunderstanding may be stemmed from my perception of the theorem)  

From here the authors explain the straightforward jump from this to that for any unsupervised learned disentangled latent space there will exist infinitely many entangled latent space with the exact same distribution.  

I do not understand why this means its no longer disentangled? Just because an entangled representation exists, does not mean the disentangled is any less valid. We can still conduct inference of the variables independently because they still follow that $p(z) = \prod_i p(z_i)$, so where does the impossibility come in?
The impossibility is referring how to learn the disentangled representations from the observed distribution or to know whether you have a disentangled representation in the first place.

Basically, an unsupervised learning agent tasked with learning a disentangled transformation of some features $\mathbf{z}$ needs to infer a set of features from the data which are not entangled, but the supplied data will always have many equally valid entangled solutions - valid from the point of view of describing the distribution of $\mathbf{z}$ accurately.

An analogy would be "I have observed the value 50, and know it is the sum of 3 numbers. What are those numbers?". Whilst the correct answer exists, and it is possible to guess it, it cannot be inferred from the supplied information. 

The part of the proof you quote shows that the multiple equivalent entangled feature sets exist, and theoretically cannot be separated from a "true" disentangled feature set on the basis of knowing the distribution. Once you accept this, it is indeed just a short hop in logic to say that the disentangled features are not learnable - there is no way for a learning system to differentiate between the entangled and disentangled features that explain the distribution, and a large (infinite) set of valid entangled features are guaranteed to exist, which will confound attempts to find a perfect solution.

It is worth noting that the impossibility refers to learning perfect solutions, and that the proof does not rule out useful or practical approximate solutions, or solutions that augment unsupervised learning by applying some additional rules or a semi-supervised approach.