# Training VAEs Under Structured Residuals

arXiv: https://arxiv.org/abs/1804.01050 · HTML: https://mlsanity.com/paper/1804.01050/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2018.

## Built on by (most cited first, 5 of 5)

- [Uncertainty Quantification via Neural Posterior Principal Components](https://mlsanity.com/paper/2309.15533/) (2023)
- [Compressed Sensing MRI Reconstruction Regularized by VAEs with Structured Image Covariance](https://mlsanity.com/paper/2210.14586/) (2022)
- [Structured Recognition for Generative Models with Explaining Away](https://mlsanity.com/paper/2209.05212/) (2022)
- [Complex variational autoencoders admit K"ahler structure](https://mlsanity.com/paper/2511.15172/) (2025)
- [It depends: Incorporating correlations for joint aleatoric and epistemic uncertainties of high-dimensional output spaces](https://mlsanity.com/paper/2608.24518/) (2026)

## Abstract (arXiv)

Variational auto-encoders (VAEs) are a popular and powerful deep generative model. Previous works on VAEs have assumed a factorized likelihood model, whereby the output uncertainty of each pixel is assumed to be independent. This approximation is clearly limited as demonstrated by observing a residual image from a VAE reconstruction, which often possess a high level of structure. This paper demonstrates a novel scheme to incorporate a structured Gaussian likelihood prediction network within the VAE that allows the residual correlations to be modeled. Our novel architecture, with minimal increase in complexity, incorporates the covariance matrix prediction within the VAE. We also propose a new mechanism for allowing structured uncertainty on color images. Furthermore, we provide a scheme for effectively training this model, and include some suggestions for improving performance in terms of efficiency or modeling longer range correlations.