# Learning ℓ1-based analysis and synthesis sparsity priors using bi-level optimization

arXiv: https://arxiv.org/abs/1401.4105 · HTML: https://mlsanity.com/paper/1401.4105/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jan 2014.

## Built on by (most cited first, 6 of 6)

- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Model-based Reconstruction with Learning: From Unsupervised to Supervised and Beyond](https://mlsanity.com/paper/2103.14528/) (2021)
- [Learning Sparsity-Promoting Regularizers using Bilevel Optimization](https://mlsanity.com/paper/2207.08939/) (2022)
- [Bilevel learning of l1-regularizers with closed-form gradients(BLORC)](https://mlsanity.com/paper/2111.10858/) (2021)
- [Controlled Learning of Pointwise Nonlinearities in Neural-Network-Like Architectures](https://mlsanity.com/paper/2408.13114/) (2024)
- [Universal Architectures for the Learning of Polyhedral Norms and Convex Regularizers](https://mlsanity.com/paper/2503.19190/) (2025)

## Abstract (arXiv)

We consider the analysis operator and synthesis dictionary learning problems based on the the ℓ1 regularized sparse representation model. We reveal the internal relations between the ℓ1-based analysis model and synthesis model. We then introduce an approach to learn both analysis operator and synthesis dictionary simultaneously by using a unified framework of bi-level optimization. Our aim is to learn a meaningful operator (dictionary) such that the minimum energy solution of the analysis (synthesis)-prior based model is as close as possible to the ground-truth. We solve the bi-level optimization problem using the implicit differentiation technique. Moreover, we demonstrate the effectiveness of our leaning approach by applying the learned analysis operator (dictionary) to the image denoising task and comparing its performance with state-of-the-art methods. Under this unified framework, we can compare the performance of the two types of priors.