Disentanglement as Identifiable Pushforward Factorisation

We characterise disentanglement for smooth generative pushforward models, such as in VAEs and GANs. For a generator/decoder $$\begin{gathered} g: \\ mathcalZ \\ to \\ mathcalX \end{gathered}$$ and factorised prior $$\begin{gathered} p(z)= \\ pro{d}_i{p}_i(z_i) \end{gathered}$$, we define disentanglement as standard statistical independence expressed over the generated manifold: the pushforward density p_\\mu = g_\\#p factorises into one–dimensional "seam" factors, each controlled by a distinct latent coordinate. We prove that p_\\mu factorises according to the SVD of $g$'s Jacobian; that disentanglement equates to two conditions on $g$ (C1-C2); and that under such conditions the seam factors are identifiable, up to permutation and sign. In the special case of Gaussian ($beta$-)VAEs, we show via an identity how diagonal posteriors promote C1-C2, in expectation, explaining why disentanglement arises modulated by $beta$. Experiments illustrate this mechanism on Gaussian data, dSprites, and CelebA.