Title: Unpicking Data at the Seams: Understanding Disentanglement in VAEs
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: disentanglement, probabilistic model, generative model, VAE, theory

Abstract: A generative latent variable model is said to be $disentangled$, when varying a single latent co-ordinate changes a single aspect of samples generated, e.g. object position or facial expression in an image. 
Related phenomena are seen in several generative paradigms, including state-of-the-art diffusion models, but disentanglement is most notably observed in Variational Autoencoders (VAEs), where oft-used $diagonal$ posterior covariances are argued to be the cause. 
We make this picture precise. From a known exact link between optimal Gaussian posteriors and decoder derivatives, we show how diagonal posteriors “lock” a decoder’s local axes so that density over the data manifold $factorises$ along $independent$ one‑dimensional seams that map to $axis-aligned$ directions in latent space. 
This gives a clear definition of disentanglement, explains why it emerges in VAEs and shows that, under stated assumptions, ground truth factors are $identifiable$ even with a symmetric prior.

TL;DR: Disentanglement = factorising the data distribution into independent factors aligned with the latent axes. Diagonal posteriors in a VAE induce the constraints necessary to achieve that.