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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?

This may be better posted in mathematics stack, however it look like Neil is on the right track – hisairnessag3 – 2019-08-12T11:13:58.463