Disentanglement as Identifiable Pushforward Factorisation
Provides a theoretical foundation for disentanglement in smooth generative models, clarifying when and why it occurs, which is important for interpretability and control in generative AI.
The paper defines disentanglement in generative models as factorisation of the pushforward density into one-dimensional factors and proves that this is equivalent to two conditions on the generator, making the factors identifiable up to permutation and sign. It also explains how diagonal posteriors in Gaussian VAEs promote these conditions, modulated by β.
We characterise disentanglement in smooth generative pushforward models, such as in VAEs and GANs. For a generator/decoder $g:Z\to X$ and factorised prior $p(z)=\prod_i p_i(z_i)$, we define disentanglement as factorisation of the pushforward density $p_μ= g_\#p$ into one-dimensional "seam" factors, where each latent dimension controls an independent generative factor of the data. We prove that $p_μ$ factorises according to the SVD of $g$'s Jacobian; that disentanglement equates to two conditions on $g$ (C1-C2); and that under those conditions the seam factors are identifiable, up to permutation and sign. In the particular case of Gaussian ($β$-)VAEs, we show via an identity how diagonal posteriors promote C1-C2, in expectation, explaining why disentanglement arises modulated by $β$. Experiments illustrate this mechanism on Gaussian data, dSprites, and CelebA.