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

**Authors:** Siheng Chen, Yonina C. Eldar, Lingxiao Zhao  
**arXiv:** [2006.01301](https://arxiv.org/abs/2006.01301)  
**Date:** 2020-06-01  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2006.01301v1/

## Key points

- Authors introduce a novel framework for denoising single or multiple noisy graph signals using interpretable graph neural networks
- Proposed "graph unrolling networks" extend the concept of algorithm unrolling to the graph domain, offering a signal processing perspective on architecture design
- Feed-forward process effectively denoises graph signals in a progressive manner by mapping each iteration of an iterative denoising algorithm to a single network layer
- Employ unsupervised learning to train the graph unrolling networks, with input noisy graph signals serving as supervision for the network
- Key feature: edge-weight-sharing graph convolution operation parameterizes edge weights using a trainable kernel function shared across all edges, allowing flexible adjustment based on different types of graph signals
- Specialized versions: Graph Unrolling Sparse Coding (GUSC) and Graph Unrolling Trend Filtering (GUTF) apply sparse coding and trend filtering techniques through network unrolling
- Extensive experiments show proposed methods outperform traditional denoising algorithms and state-of-the-art graph neural networks, achieving significantly lower normalized mean square error compared to graph Laplacian denoising and graph wavelets methods

## Summary

In their paper titled "Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising," authors Siheng Chen, Yonina C. Eldar, and Lingxiao Zhao introduce a novel framework for denoising single or multiple noisy graph signals using interpretable graph neural networks. The proposed <kw>graph unrolling networks</kw> extend the concept of algorithm unrolling to the graph domain, offering a signal processing perspective on architecture design. By mapping each iteration of an iterative denoising algorithm to a single network layer, the feed-forward process effectively denoises graph signals in a progressive manner. The authors employ <kw>unsupervised learning</kw> to train the graph unrolling networks, with input noisy graph signals serving as supervision for the network. This approach leverages the learning capabilities of neural networks to adaptively capture relevant priors from input noisy signals, eliminating the need for manual selection of signal priors. A key feature of these networks is the <kw>edge-weight-sharing graph convolution operation</kw>, which parameterizes edge weights using a trainable kernel function shared across all edges. This permutation-equivariant convolution allows for flexible adjustment of edge weights based on different types of graph signals. Furthermore, the authors explore two specialized versions of these networks: <kw>Graph Unrolling Sparse Coding (GUSC)</kw> and <kw>Graph Unrolling Trend Filtering (GUTF)</kw>, which apply sparse coding and trend filtering techniques through network unrolling. Extensive experiments conducted on real-world and simulated datasets demonstrate that their proposed methods outperform traditional denoising algorithms and state-of-the-art graph neural networks. For instance, when denoising a single smooth graph signal, the normalized mean square error achieved by their networks is approximately 40% and 60% lower than that obtained by <kw>graph Laplacian denoising</kw> and <kw>graph wavelets methods</kw> respectively. Overall, this research introduces an innovative approach to denoising graph signals through interpretable neural networks, showcasing superior performance compared to existing methods in various experimental settings.