# Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising

arXiv: https://arxiv.org/abs/2006.01301 · HTML: https://mlsanity.com/paper/2006.01301/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 30 of 33)

- [Elastic Graph Neural Networks](https://mlsanity.com/paper/2107.06996/) (2021)
- [Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry](https://mlsanity.com/paper/2503.01822/) (2025)
- [Equivariant Hypergraph Diffusion Neural Operators](https://mlsanity.com/paper/2207.06680/) (2022)
- [RU-Net: Regularized Unrolling Network for Scene Graph Generation](https://mlsanity.com/paper/2205.01297/) (2022)
- [LazyGNN: Large-Scale Graph Neural Networks via Lazy Propagation](https://mlsanity.com/paper/2302.01503/) (2023)
- [Untrained Graph Neural Networks for Denoising](https://mlsanity.com/paper/2109.11700/) (2021)
- [Towards Data-centric Graph Machine Learning: Review and Outlook](https://mlsanity.com/paper/2309.10979/) (2023)
- [Graph Signal Restoration Using Nested Deep Algorithm Unrolling](https://mlsanity.com/paper/2106.15910/) (2021)
- [Interpretability of Machine Learning: Recent Advances and Future Prospects](https://mlsanity.com/paper/2305.00537/) (2023)
- [Graph Denoising with Framelet Regularizer](https://mlsanity.com/paper/2111.03264/) (2021)
- [Graph-Aware Diffusion for Signal Generation](https://mlsanity.com/paper/2510.05036/) (2025)
- [Robust Graph Representation Learning for Local Corruption Recovery](https://mlsanity.com/paper/2202.04936/) (2022)
- [Robust PCA Unrolling Network for Super-resolution Vessel Extraction in X-ray Coronary Angiography](https://mlsanity.com/paper/2204.08466/) (2022)
- [Stochastic Unrolled Federated Learning](https://mlsanity.com/paper/2305.15371/) (2023)
- [Graph-based Algorithm Unfolding for Energy-aware Power Allocation in Wireless Networks](https://mlsanity.com/paper/2201.11799/) (2022)
- [How Powerful is Implicit Denoising in Graph Neural Networks](https://mlsanity.com/paper/2209.14514/) (2022)
- [ASGNN: Graph Neural Networks with Adaptive Structure](https://mlsanity.com/paper/2210.01002/) (2022)
- [MuseGNN: Forming Scalable, Convergent GNN Layers that Minimize a Sampling-Based Energy](https://mlsanity.com/paper/2310.12457/) (2023)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Unrolled algorithms for group synchronization](https://mlsanity.com/paper/2207.09418/) (2022)
- [Learn What You Need in Personalized Federated Learning](https://mlsanity.com/paper/2401.08327/) (2024)
- [Scalable Weibull Graph Attention Autoencoder for Modeling Document Networks](https://mlsanity.com/paper/2410.09696/) (2024)
- [Enhancing Node-Level Graph Domain Adaptation by Alleviating Local Dependency](https://mlsanity.com/paper/2512.13149/) (2025)
- [DPC: Unsupervised Deep Point Correspondence via Cross and Self Construction](https://mlsanity.com/paper/2110.08636/) (2021)
- [Hyper-GST: Predict Metro Passenger Flow Incorporating GraphSAGE, Hypergraph, Social-meaningful Edge Weights and Temporal Exploitation](https://mlsanity.com/paper/2211.04988/) (2022)
- [Graph Adversarial Diffusion Convolution](https://mlsanity.com/paper/2406.02059/) (2024)
- [Joint Data Inpainting and Graph Learning via Unrolled Neural Networks](https://mlsanity.com/paper/2407.11429/) (2024)
- [A Review of Graph-Powered Data Quality Applications for IoT Monitoring Sensor Networks](https://mlsanity.com/paper/2410.21006/) (2024)
- [LASE: Learned Adjacency Spectral Embeddings](https://mlsanity.com/paper/2412.17734/) (2024)
- [Unrolling Dynamic Programming via Graph Filters](https://mlsanity.com/paper/2507.21705/) (2025)

## Abstract (arXiv)

We propose an interpretable graph neural network framework to denoise single or multiple noisy graph signals. The proposed graph unrolling networks expand algorithm unrolling to the graph domain and provide an interpretation of the architecture design from a signal processing perspective. We unroll an iterative denoising algorithm by mapping each iteration into a single network layer where the feed-forward process is equivalent to iteratively denoising graph signals. We train the graph unrolling networks through unsupervised learning, where the input noisy graph signals are used to supervise the networks. By leveraging the learning ability of neural networks, we adaptively capture appropriate priors from input noisy graph signals, instead of manually choosing signal priors. A core component of graph unrolling networks is the edge-weight-sharing graph convolution operation, which parameterizes each edge weight by a trainable kernel function where the trainable parameters are shared by all the edges. The proposed convolution is permutation-equivariant and can flexibly adjust the edge weights to various graph signals. We then consider two special cases of this class of networks, graph unrolling sparse coding (GUSC) and graph unrolling trend filtering (GUTF), by unrolling sparse coding and trend filtering, respectively. To validate the proposed methods, we conduct extensive experiments on both real-world datasets and simulated datasets, and demonstrate that our methods have smaller denoising errors than conventional denoising algorithms and state-of-the-art graph neural networks. For denoising a single smooth graph signal, the normalized mean square error of the proposed networks is around 40% and 60% lower than that of graph Laplacian denoising and graph wavelets, respectively.