# β-VAEs as Effective Theories: Tolerance-Dependent Dimension

arXiv: https://arxiv.org/abs/2608.10599 · HTML: https://mlsanity.com/paper/2608.10599/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 19 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Don't Blame the ELBO! A Linear VAE Perspective on Posterior Collapse](https://mlsanity.com/paper/1911.02469/) (2019)
- [Why does deep and cheap learning work so well?](https://mlsanity.com/paper/1608.08225/) (2016)
- [An exact mapping between the Variational Renormalization Group and Deep Learning](https://mlsanity.com/paper/1410.3831/) (2014)
- [The Usual Suspects? Reassessing Blame for VAE Posterior Collapse](https://mlsanity.com/paper/1912.10702/) (2019)
- [Latent Spectroscopy: Posterior Collapse as a Feature](https://mlsanity.com/paper/2605.22691/) (2026)

## Abstract (arXiv)

In a β-VAE, increasing the regularization strength acts as a spectral cutoff by collapsing low-utility latent coordinates. In the linear Gaussian VAE, the collapse order matches the ranking of reconstruction utilities exactly, because both are set by the PCA spectrum. We ask which parts of this picture survive in fully connected nonlinear VAEs trained on WorldClim. We find that nonlinear interactions shift and broaden collapse onsets, so thresholds no longer coincide exactly with utilities. However, the common ordering is preserved over the resolved ranks, so the spectral cutoff still acts as a utility cutoff and the effective-description logic carries through. The resulting effective-dimension curves reveal a head--tail tradeoff: increasing depth concentrates utility into the first few coordinates but worsens tail fidelity.