Disentanglement as Identifiable Pushforward Factorisation
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913021833510912 |
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| author | Allen, Carl |
| author_facet | Allen, Carl |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22559 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Disentanglement as Identifiable Pushforward Factorisation Allen, Carl Machine Learning Artificial Intelligence 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. |
| title | Disentanglement as Identifiable Pushforward Factorisation |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2410.22559 |