# A Testable Certificate for Constant Collapse in Teacher-Guided VAEs

arXiv: https://arxiv.org/abs/2605.05813 · HTML: https://mlsanity.com/paper/2605.05813/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 8 references that are in the map.
- First posted on arXiv 7 May 2026.

## Builds on (most cited first, 5 of 5)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders](https://mlsanity.com/paper/1611.02648/) (2016)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Lagging Inference Networks and Posterior Collapse in Variational Autoencoders](https://mlsanity.com/paper/1901.05534/) (2019)

## Built on by (most cited first, 1 of 1)

- [RST: certifying and constructing prescribed information in variational autoencoders](https://mlsanity.com/paper/2605.18224/) (2026)

## Abstract (arXiv)

Posterior collapse in variational autoencoders is often diagnosed by its symptoms: a small KL term, a strong decoder, or weak use of the latent code. These signals are useful, but they do not define a collapse boundary. We study a concrete failure mode, input-independent constant collapse, and show that this case admits an exact threshold. For any fixed nonconstant teacher distribution (T(·mid x)), the best constant student is the dataset-average teacher distribution, and its alignment cost is the teacher mutual information (IT(X;T)). Therefore, if a strictly latent-only raw witness achieves alignment loss below this value, with a safety margin, the witness cannot be constant in the input. This identity turns a qualitative failure mode into a measurable one. In CIFAR-100 experiments with per-seed teacher search, full training stays on the certified side of the boundary, removing alignment drives the raw witness into the constant-student regime, and restarting from a collapsed checkpoint with alignment enabled restores the certificate. Tiny-ImageNet-200 fixed-target runs show the same prevention--collapse--rescue pattern across three independently searched teachers. Standard VAE-style baselines, including methods that preserve reconstruction quality or post-hoc predictability, remain negative under the raw certificate. The guarantee is intentionally narrow: it certifies that the matched nonconstant teacher-relative variation passes through the latent pathway, rather than claiming that all forms of posterior collapse have been ruled out.