# Memory-efficient deep end-to-end posterior network (DEEPEN) for inverse problems

arXiv: https://arxiv.org/abs/2402.05422 · HTML: https://mlsanity.com/paper/2402.05422/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 12 references that are in the map.
- First posted on arXiv 8 Feb 2024.

## Builds on (most cited first, 7 of 7)

- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Memory-efficient model-based deep learning with convergence and robustness guarantees](https://mlsanity.com/paper/2206.04797/) (2022)

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

- [Deep End-to-End Posterior ENergy (DEEPEN) for image recovery](https://mlsanity.com/paper/2503.17244/) (2025)

## Abstract (arXiv)

End-to-End (E2E) unrolled optimization frameworks show promise for Magnetic Resonance (MR) image recovery, but suffer from high memory usage during training. In addition, these deterministic approaches do not offer opportunities for sampling from the posterior distribution. In this paper, we introduce a memory-efficient approach for E2E learning of the posterior distribution. We represent this distribution as the combination of a data-consistency-induced likelihood term and an energy model for the prior, parameterized by a Convolutional Neural Network (CNN). The CNN weights are learned from training data in an E2E fashion using maximum likelihood optimization. The learned model enables the recovery of images from undersampled measurements using the Maximum A Posteriori (MAP) optimization. In addition, the posterior model can be sampled to derive uncertainty maps about the reconstruction. Experiments on parallel MR image reconstruction show that our approach performs comparable to the memory-intensive E2E unrolled algorithm, performs better than its memory-efficient counterpart, and can provide uncertainty maps. Our framework paves the way towards MR image reconstruction in 3D and higher dimensions