Disentanglement as Identifiable Pushforward Factorisation

2024/10/29 by Carl Allen, Allen, Carl
Computer Science · Neuroscience · #Artificial Intelligence (cs.AI) #Embodied and Extended Cognition #FOS: Computer and information sciences #Face Recognition and Perception #Generative Adversarial Networks and Image Synthesis #Machine Learning (cs.LG) #Machine Learning (stat.ML)

paper · pdf · doi:10.48550/arxiv.2410.22559

openalex publication_date 2024/10/29 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

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)=∏_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.

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