# Variational Sparse Paired Autoencoders (vsPAIR) for Inverse Problems and Uncertainty Quantification

arXiv: https://arxiv.org/abs/2602.02948 · HTML: https://mlsanity.com/paper/2602.02948/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 43 of its 86 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 30 of 43)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2201.11793/) (2022)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [The Little Engine that Could: Regularization by Denoising (RED)](https://mlsanity.com/paper/1611.02862/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Image reconstruction by domain transform manifold learning](https://mlsanity.com/paper/1704.08841/) (2017)
- [Discrete Variational Autoencoders](https://mlsanity.com/paper/1609.02200/) (2016)
- [Generalized Denoising Auto-Encoders as Generative Models](https://mlsanity.com/paper/1305.6663/) (2013)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [Deep Probabilistic Imaging: Uncertainty Quantification and Multi-modal Solution Characterization for Computational Imaging](https://mlsanity.com/paper/2010.14462/) (2020)
- [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020)
- [A review of uncertainty quantification in medical image analysis: probabilistic and non-probabilistic methods](https://mlsanity.com/paper/2310.06873/) (2023)
- [Solving Bayesian Inverse Problems via Variational Autoencoders](https://mlsanity.com/paper/1912.04212/) (2019)
- [Conditional Variational Autoencoder for Learned Image Reconstruction](https://mlsanity.com/paper/2110.11681/) (2021)
- [Conditional Normalizing Flows for Low-Dose Computed Tomography Image Reconstruction](https://mlsanity.com/paper/2006.06270/) (2020)

## Abstract (arXiv)

Inverse problems are fundamental to many scientific and engineering disciplines; they arise when one seeks to reconstruct hidden, underlying quantities from noisy measurements. Many applications demand not just point estimates but interpretable uncertainty. Providing fast inference alongside uncertainty estimates remains challenging yet desirable in numerous applications. We propose the Variational Sparse Paired Autoencoder (vsPAIR) to address this challenge. The architecture pairs a standard VAE encoding observations with a sparse VAE encoding quantities of interest (QoI), connected through a learned latent mapping. The variational structure enables uncertainty estimation, the paired architecture encourages interpretability by anchoring QoI representations to clean data, and sparse encodings provide structure by concentrating information into identifiable factors rather than diffusing across all dimensions. To validate the effectiveness of our proposed architecture, we conduct experiments on blind inpainting, computed tomography (CT), and initial-condition inference for the heat equation, demonstrating that vsPAIR is a capable inverse problem solver that can provide interpretable and structured uncertainty estimates.