I Don't Need $\mathbf{u}$: Identifiable Non-Linear ICA Without Side Information

Recently there has been a renaissance in identifiability results in deep generative models, not least for non-linear ICA. For i.i.d. data, prior works have assumed access to a sufficiently-informative auxiliary set of observations, denoted $\mathbf{u}$. We show here how identifiability can be obtained in the absence of this side-information. Previous methods have had to make strong assumptions in order to obtain identifiable models. Here we obtain empirically identifiable models under a much looser set of constraints. In particular, we focus on generative models which perform clustering in their latent space -- a model structure which matches previous identifiable models, but with the learnt clustering providing a synthetic form of auxiliary information. We evaluate our proposals, including via statistical tests, and find that the learned clusterings function effectively: deep generative models with latent clusterings are empirically identifiable, to the same degree as models which rely on side information.