# Graph Signal Restoration Using Nested Deep Algorithm Unrolling

arXiv: https://arxiv.org/abs/2106.15910 · HTML: https://mlsanity.com/paper/2106.15910/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 64 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing](https://mlsanity.com/paper/1912.10557/) (2019)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Deep Plug-and-Play Super-Resolution for Arbitrary Blur Kernels](https://mlsanity.com/paper/1903.12529/) (2019)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Tuning-free Plug-and-Play Proximal Algorithm for Inverse Imaging Problems](https://mlsanity.com/paper/2002.09611/) (2020)
- [Discrete Signal Processing on Graphs: Sampling Theory](https://mlsanity.com/paper/1503.05432/) (2015)
- [Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising](https://mlsanity.com/paper/2006.01301/) (2020)
- [Signal Recovery on Graphs: Variation Minimization](https://mlsanity.com/paper/1411.7414/) (2014)
- [Deep Unfolding of a Proximal Interior Point Method for Image Restoration](https://mlsanity.com/paper/1812.04276/) (2018)
- [Performance Analysis of Plug-and-Play ADMM: A Graph Signal Processing Perspective](https://mlsanity.com/paper/1809.00020/) (2018)
- [Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison (Extended Cut)](https://mlsanity.com/paper/2006.11220/) (2020)
- [Bilateral Filter: Graph Spectral Interpretation and Extensions](https://mlsanity.com/paper/1303.2685/) (2013)
- [Dynamically Unfolding Recurrent Restorer: A Moving Endpoint Control Method for Image Restoration](https://mlsanity.com/paper/1805.07709/) (2018)
- [Generalized Sampling on Graphs With Subspace and Smoothness Priors](https://mlsanity.com/paper/1905.04441/) (2019)
- [Fast Singular Value Shrinkage with Chebyshev Polynomial Approximation Based on Signal Sparsity](https://mlsanity.com/paper/1705.07112/) (2017)

## Built on by (most cited first, 13 of 13)

- [GSP-KalmanNet: Tracking Graph Signals via Neural-Aided Kalman Filtering](https://mlsanity.com/paper/2311.16602/) (2023)
- [Graph-Aware Diffusion for Signal Generation](https://mlsanity.com/paper/2510.05036/) (2025)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Learning Graph ARMA Processes from Time-Vertex Spectra](https://mlsanity.com/paper/2302.06887/) (2023)
- [Unrolled Graph Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2509.17156/) (2025)
- [Graph Filter Transfer via Probability Density Ratio Weighting](https://mlsanity.com/paper/2210.14633/) (2022)
- [Deep Unfolding-based Weighted Averaging for Federated Learning in Heterogeneous Environments](https://mlsanity.com/paper/2212.12191/) (2022)
- [Locally Stationary Graph Processes](https://mlsanity.com/paper/2309.01657/) (2023)
- [LASE: Learned Adjacency Spectral Embeddings](https://mlsanity.com/paper/2412.17734/) (2024)
- [Efficient Learning of Balanced Signed Graphs via Sparse Linear Programming](https://mlsanity.com/paper/2506.01826/) (2025)
- [Sampling Method for Generalized Graph Signals with Pre-selected Vertices via DC Optimization](https://mlsanity.com/paper/2509.14836/) (2025)
- [Unrolled Neural Networks for Constrained Optimization](https://mlsanity.com/paper/2601.17274/) (2026)
- [Low-rank Updates in Slowly Time-varying Graphs for Spatial-Temporal Signal Interpolation](https://mlsanity.com/paper/2606.24011/) (2026)

## Abstract (arXiv)

Graph signal processing is a ubiquitous task in many applications such as sensor, social, transportation and brain networks, point cloud processing, and graph neural networks. Often, graph signals are corrupted in the sensing process, thus requiring restoration. In this paper, we propose two graph signal restoration methods based on deep algorithm unrolling (DAU). First, we present a graph signal denoiser by unrolling iterations of the alternating direction method of multiplier (ADMM). We then suggest a general restoration method for linear degradation by unrolling iterations of Plug-and-Play ADMM (PnP-ADMM). In the second approach, the unrolled ADMM-based denoiser is incorporated as a submodule, leading to a nested DAU structure. The parameters in the proposed denoising/restoration methods are trainable in an end-to-end manner. Our approach is interpretable and keeps the number of parameters small since we only tune graph-independent regularization parameters. We overcome two main challenges in existing graph signal restoration methods: 1) limited performance of convex optimization algorithms due to fixed parameters which are often determined manually. 2) large number of parameters of graph neural networks that result in difficulty of training. Several experiments for graph signal denoising and interpolation are performed on synthetic and real-world data. The proposed methods show performance improvements over several existing techniques in terms of root mean squared error in both tasks.