# Disentanglement as Identifiable Pushforward Factorisation

arXiv: https://arxiv.org/abs/2410.22559 · HTML: https://mlsanity.com/paper/2410.22559/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 41 references that are in the map.
- First posted on arXiv 29 Oct 2024.

## Builds on (most cited first, 25 of 25)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [Don't Blame the ELBO! A Linear VAE Perspective on Posterior Collapse](https://mlsanity.com/paper/1911.02469/) (2019)
- [DiffuseVAE: Efficient, Controllable and High-Fidelity Generation from Low-Dimensional Latents](https://mlsanity.com/paper/2201.00308/) (2022)
- [Independent mechanism analysis, a new concept?](https://mlsanity.com/paper/2106.05200/) (2021)
- [The Hessian Penalty: A Weak Prior for Unsupervised Disentanglement](https://mlsanity.com/paper/2008.10599/) (2020)
- [Taming VAEs](https://mlsanity.com/paper/1810.00597/) (2018)
- [Variational Autoencoders Pursue PCA Directions (by Accident)](https://mlsanity.com/paper/1812.06775/) (2018)
- [Additive Decoders for Latent Variables Identification and Cartesian-Product Extrapolation](https://mlsanity.com/paper/2307.02598/) (2023)
- [Unsupervised Representation Learning from Pre-trained Diffusion Probabilistic Models](https://mlsanity.com/paper/2212.12990/) (2022)
- [Function Classes for Identifiable Nonlinear Independent Component Analysis](https://mlsanity.com/paper/2208.06406/) (2022)
- [Provably Learning Object-Centric Representations](https://mlsanity.com/paper/2305.14229/) (2023)
- [Orthogonal Jacobian Regularization for Unsupervised Disentanglement in Image Generation](https://mlsanity.com/paper/2108.07668/) (2021)
- [On Implicit Regularization in β-VAEs](https://mlsanity.com/paper/2002.00041/) (2020)
- [DisDiff: Unsupervised Disentanglement of Diffusion Probabilistic Models](https://mlsanity.com/paper/2301.13721/) (2023)
- [Embrace the Gap: VAEs Perform Independent Mechanism Analysis](https://mlsanity.com/paper/2206.02416/) (2022)
- [Regularized linear autoencoders recover the principal components, eventually](https://mlsanity.com/paper/2007.06731/) (2020)
- [A Geometric Perspective on Variational Autoencoders](https://mlsanity.com/paper/2209.07370/) (2022)
- [Variational autoencoders in the presence of low-dimensional data: landscape and implicit bias](https://mlsanity.com/paper/2112.06868/) (2021)
- [Demystifying μ](https://mlsanity.com/paper/2401.01096/) (2024)
- [Variational Classification](https://mlsanity.com/paper/2305.10406/) (2023)

## Abstract (arXiv)

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.