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. 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 whose 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 further 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.

中文翻译：

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图展开网络：用于图信号去噪的可解释神经网络


我们提出了一种可解释的图神经网络框架来对单个或多个噪声图信号进行去噪。所提出的图展开网络将算法展开到图域，并从信号处理的角度提供了架构设计的解释。我们通过将每次迭代映射到单个网络层来展开迭代去噪算法，其中前馈过程相当于对图信号进行迭代去噪。我们通过无监督学习来训练图展开网络，其中输入的噪声图信号用于监督网络。通过利用神经网络的学习能力，我们从输入的噪声图信号中自适应地捕获适当的先验。图展开网络的核心组件是边权重共享图卷积运算，它通过可训练的核函数对每个边权重进行参数化，该核函数的可训练参数由所有边共享。所提出的卷积是排列等变的，可以灵活地调整边缘权重以适应各种图信号。我们进一步通过分别展开稀疏编码和趋势过滤来考虑此类网络的两种特殊情况，图展开稀疏编码（GUSC）和图展开趋势过滤（GUTF）。为了验证所提出的方法，我们在现实数据集和模拟数据集上进行了广泛的实验，并证明我们的方法比传统的去噪算法和最先进的图神经网络具有更小的去噪误差。对于单个平滑图信号的去噪，所提出的网络的归一化均方误差分别比图拉普拉斯去噪和图小波的归一化均方误差低约 40% 和 60%。