Disentanglement as Identifiable Pushforward Factorisation

We characterise disentanglement in smooth generative pushforward models, such as in VAEs and GANs. For a generator/decoder g:Z→ X and factorised prior p(z)=prodi pi(zi), we define disentanglement as factorisation of the pushforward density pμ= g_#p into one-dimensional "seam" factors, where each latent dimension controls an independent generative factor of the data. We prove that pμ 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.