# Learning to solve TV regularized problems with unrolled algorithms

arXiv: https://arxiv.org/abs/2010.09545 · HTML: https://mlsanity.com/paper/2010.09545/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Oct 2020.

## Built on by (most cited first, 8 of 8)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Efficient and Modular Implicit Differentiation](https://mlsanity.com/paper/2105.15183/) (2021)
- [Implicit differentiation for fast hyperparameter selection in non-smooth convex learning](https://mlsanity.com/paper/2105.01637/) (2021)
- [Total Variation Optimization Layers for Computer Vision](https://mlsanity.com/paper/2204.03643/) (2022)
- [Deep Unrolling Networks with Recurrent Momentum Acceleration for Nonlinear Inverse Problems](https://mlsanity.com/paper/2307.16120/) (2023)
- [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning](https://mlsanity.com/paper/2407.13666/) (2024)
- [Double Momentum Method for Lower-Level Constrained Bilevel Optimization](https://mlsanity.com/paper/2406.17386/) (2024)
- [Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms](https://mlsanity.com/paper/2411.13276/) (2024)

## Abstract (arXiv)

Total Variation (TV) is a popular regularization strategy that promotes piece-wise constant signals by constraining the ℓ1-norm of the first order derivative of the estimated signal. The resulting optimization problem is usually solved using iterative algorithms such as proximal gradient descent, primal-dual algorithms or ADMM. However, such methods can require a very large number of iterations to converge to a suitable solution. In this paper, we accelerate such iterative algorithms by unfolding proximal gradient descent solvers in order to learn their parameters for 1D TV regularized problems. While this could be done using the synthesis formulation, we demonstrate that this leads to slower performances. The main difficulty in applying such methods in the analysis formulation lies in proposing a way to compute the derivatives through the proximal operator. As our main contribution, we develop and characterize two approaches to do so, describe their benefits and limitations, and discuss the regime where they can actually improve over iterative procedures. We validate those findings with experiments on synthetic and real data.