# Hybrid ISTA: Unfolding ISTA With Convergence Guarantees Using Free-Form Deep Neural Networks

arXiv: https://arxiv.org/abs/2204.11640 · HTML: https://mlsanity.com/paper/2204.11640/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 61 references that are in the map.
- First posted on arXiv 25 Apr 2022.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Architecture Search with Reinforcement Learning](https://mlsanity.com/paper/1611.01578/) (2016)
- [Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds](https://mlsanity.com/paper/1808.10038/) (2018)
- [Learned D-AMP: Principled Neural Network based Compressive Image Recovery](https://mlsanity.com/paper/1704.06625/) (2017)
- [Gap Safe screening rules for sparsity enforcing penalties](https://mlsanity.com/paper/1611.05780/) (2016)
- [Celer: a Fast Solver for the Lasso with Dual Extrapolation](https://mlsanity.com/paper/1802.07481/) (2018)
- [DR2-Net: Deep Residual Reconstruction Network for Image Compressive Sensing](https://mlsanity.com/paper/1702.05743/) (2017)
- [Maximal Sparsity with Deep Networks?](https://mlsanity.com/paper/1605.01636/) (2016)
- [Splitting methods with variable metric for KL functions](https://mlsanity.com/paper/1405.1357/) (2014)
- [Learned Convolutional Sparse Coding](https://mlsanity.com/paper/1711.00328/) (2017)
- [Rethinking the CSC Model for Natural Images](https://mlsanity.com/paper/1909.05742/) (2019)
- [Differentiable Linearized ADMM](https://mlsanity.com/paper/1905.06179/) (2019)
- [A Tight Bound of Hard Thresholding](https://mlsanity.com/paper/1605.01656/) (2016)
- [Linear Convergence of Stochastic Iterative Greedy Algorithms with Sparse Constraints](https://mlsanity.com/paper/1407.0088/) (2014)
- [Tradeoffs between Convergence Speed and Reconstruction Accuracy in Inverse Problems](https://mlsanity.com/paper/1605.09232/) (2016)
- [Understanding Trainable Sparse Coding via Matrix Factorization](https://mlsanity.com/paper/1609.00285/) (2016)
- [Local Convergence Properties of SAGA/Prox-SVRG and Acceleration](https://mlsanity.com/paper/1802.02554/) (2018)
- [Learning A Task-Specific Deep Architecture For Clustering](https://mlsanity.com/paper/1509.00151/) (2015)
- [ReconNet: Non-Iterative Reconstruction of Images from Compressively Sensed Random Measurements](https://mlsanity.com/paper/1601.06892/) (2016)
- [Frank-Wolfe Network: An Interpretable Deep Structure for Non-Sparse Coding](https://mlsanity.com/paper/1802.10252/) (2018)
- [Extragradient Method in Optimization: Convergence and Complexity](https://mlsanity.com/paper/1609.08177/) (2016)

## Built on by (most cited first, 4 of 4)

- [Uncertainty quantification for learned ISTA](https://mlsanity.com/paper/2309.07982/) (2023)
- [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning](https://mlsanity.com/paper/2407.13666/) (2024)
- [MB-RACS: Measurement-Bounds-based Rate-Adaptive Image Compressed Sensing Network](https://mlsanity.com/paper/2402.16855/) (2024)
- [LargeMvC-Net: Anchor-based Deep Unfolding Network for Large-scale Multi-view Clustering](https://mlsanity.com/paper/2507.20980/) (2025)

## Abstract (arXiv)

It is promising to solve linear inverse problems by unfolding iterative algorithms (e.g., iterative shrinkage thresholding algorithm (ISTA)) as deep neural networks (DNNs) with learnable parameters. However, existing ISTA-based unfolded algorithms restrict the network architectures for iterative updates with the partial weight coupling structure to guarantee convergence. In this paper, we propose hybrid ISTA to unfold ISTA with both pre-computed and learned parameters by incorporating free-form DNNs (i.e., DNNs with arbitrary feasible and reasonable network architectures), while ensuring theoretical convergence. We first develop HCISTA to improve the efficiency and flexibility of classical ISTA (with pre-computed parameters) without compromising the convergence rate in theory. Furthermore, the DNN-based hybrid algorithm is generalized to popular variants of learned ISTA, dubbed HLISTA, to enable a free architecture of learned parameters with a guarantee of linear convergence. To our best knowledge, this paper is the first to provide a convergence-provable framework that enables free-form DNNs in ISTA-based unfolded algorithms. This framework is general to endow arbitrary DNNs for solving linear inverse problems with convergence guarantees. Extensive experiments demonstrate that hybrid ISTA can reduce the reconstruction error with an improved convergence rate in the tasks of sparse recovery and compressive sensing.