Graph Signal Denoising Using Regularization by Denoising and Its Parameter Estimation

In this paper, we propose an interpretable denoising method for graph signals using regularization by denoising (RED). RED is a technique developed for image restoration that uses an efficient (and sometimes black-box) denoiser in the regularization term of the optimization problem. By using RED, optimization problems can be designed with the explicit use of the denoiser, and the gradient of the regularization term can be easily computed under mild conditions. We adapt RED for denoising of graph signals beyond image processing. We show that many graph signal denoisers, including graph neural networks, theoretically or practically satisfy the conditions for RED. We also study the effectiveness of RED from a graph filter perspective. Furthermore, we propose supervised and unsupervised parameter estimation methods based on deep algorithm unrolling. These methods aim to enhance the algorithm applicability, particularly in the unsupervised setting. Denoising experiments for synthetic and real-world datasets show that our proposed method improves signal denoising accuracy in mean squared error compared to existing graph signal denoising methods.

Algorithm Unrolling-based Denoising of Multimodal Graph Signals

We propose a denoising method of multimodal graph signals by iteratively solving signal restoration and graph learning problems. Many complex-structured data, i.e., those on sensor networks, can capture multiple modalities at each measurement point, referred to as modalities. They are also assumed to have an underlying structure or correlations in modality as well as space. Such multimodal data are regarded as graph signals on a twofold graph and they are often corrupted by noise. Furthermore, their spatial/modality relationships are not always given a priori: We need to estimate twofold graphs during a denoising algorithm. In this paper, we consider a signal denoising method on twofold graphs, where graphs are learned simultaneously. We formulate an optimization problem for that and parameters in an iterative algorithm are learned from training data by unrolling the iteration with deep algorithm unrolling. Experimental results on synthetic and real-world data demonstrate that the proposed method outperforms existing model- and deep learning-based graph signal denoising methods.

Multiscale Graph Construction Using Non-local Cluster Features

This paper presents a multiscale graph construction method using both graph and signal features. Multiscale graph is a hierarchical representation of the graph, where a node at each level indicates a cluster in a finer resolution. To obtain the hierarchical clusters, existing methods often use graph clustering; however, they may ignore signal variations. As a result, these methods could fail to detect the clusters having similar features on nodes. In this paper, we consider graph and node-wise features simultaneously for multiscale clustering of a graph. With given clusters of the graph, the clusters are merged hierarchically in three steps: 1) Feature vectors in the clusters are extracted. 2) Similarities among cluster features are calculated using optimal transport. 3) A variable $k$-nearest neighbor graph (V$k$NNG) is constructed and graph spectral clustering is applied to the V$k$NNG to obtain clusters at a coarser scale. Additionally, the multiscale graph in this paper has \textit{non-local} characteristics: Nodes with similar features are merged even if they are spatially separated. In experiments on multiscale image and point cloud segmentation, we demonstrate the effectiveness of the proposed method.