# Unsupervised Deep Basis Pursuit: Learning inverse problems without ground-truth data

arXiv: https://arxiv.org/abs/1910.13110 · HTML: https://mlsanity.com/paper/1910.13110/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2019.

## Built on by (most cited first, 10 of 10)

- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [A review and experimental evaluation of deep learning methods for MRI reconstruction](https://mlsanity.com/paper/2109.08618/) (2021)
- [Self-supervised Deep Unrolled Reconstruction Using Regularization by Denoising](https://mlsanity.com/paper/2205.03519/) (2022)
- [MRI Super-Resolution with Deep Learning: A Comprehensive Survey](https://mlsanity.com/paper/2511.16854/) (2025)
- [High Fidelity Deep Learning-based MRI Reconstruction with Instance-wise Discriminative Feature Matching Loss](https://mlsanity.com/paper/2108.12460/) (2021)
- [Memory-efficient Learning for High-Dimensional MRI Reconstruction](https://mlsanity.com/paper/2103.04003/) (2021)
- [Robust multi-coil MRI reconstruction via self-supervised denoising](https://mlsanity.com/paper/2411.12919/) (2024)
- [Benchmarking Self-Supervised Learning Methods for Accelerated MRI Reconstruction](https://mlsanity.com/paper/2502.14009/) (2025)
- [A Self-Supervised Approach to Reconstruction in Sparse X-Ray Computed Tomography](https://mlsanity.com/paper/2211.00002/) (2022)
- [HyDRA: Hybrid Denoising Regularization for Measurement-Only DEQ Training](https://mlsanity.com/paper/2601.01228/) (2026)

## Abstract (arXiv)

Basis pursuit is a compressed sensing optimization in which the l1-norm is minimized subject to model error constraints. Here we use a deep neural network prior instead of l1-regularization. Using known noise statistics, we jointly learn the prior and reconstruct images without access to ground-truth data. During training, we use alternating minimization across an unrolled iterative network and jointly solve for the neural network weights and training set image reconstructions. At inference, we fix the weights and pass the measurements through the network. We compare reconstruction performance between unsupervised and supervised (i.e. with ground-truth) methods. We hypothesize this technique could be used to learn reconstruction when ground-truth data are unavailable, such as in high-resolution dynamic MRI.