# Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems

arXiv: https://arxiv.org/abs/1605.09232 · HTML: https://mlsanity.com/paper/1605.09232/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2016.

## Built on by (most cited first, 12 of 12)

- [Learning to Optimize: A Primer and A Benchmark](https://mlsanity.com/paper/2103.12828/) (2021)
- [Hyperparameter Tuning is All You Need for LISTA](https://mlsanity.com/paper/2110.15900/) (2021)
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- [Stable and Interpretable Unrolled Dictionary Learning](https://mlsanity.com/paper/2106.00058/) (2021)
- [Stochastic Unrolled Federated Learning](https://mlsanity.com/paper/2305.15371/) (2023)
- [Algorithm Unrolling for Massive Access via Deep Neural Network with Theoretical Guarantee](https://mlsanity.com/paper/2106.10426/) (2021)
- [Hybrid ISTA: Unfolding ISTA With Convergence Guarantees Using Free-Form Deep Neural Networks](https://mlsanity.com/paper/2204.11640/) (2022)
- [Unrolling PALM for sparse semi-blind source separation](https://mlsanity.com/paper/2112.05694/) (2021)
- [A Design Space Study for LISTA and Beyond](https://mlsanity.com/paper/2104.04110/) (2021)
- [Variational Learning ISTA](https://mlsanity.com/paper/2407.06646/) (2024)
- [Painless step size adaptation for SGD](https://mlsanity.com/paper/2102.00853/) (2021)
- [Learned Interpretable Residual Extragradient ISTA for Sparse Coding](https://mlsanity.com/paper/2106.11970/) (2021)

## Abstract (arXiv)

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).