First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper proposes an approach for constructing a linear wavelet transform on weighted graphs based on the lifting scheme, which has a number of favourable properties: 1) linear in memory and time with the size of the graph, 2) adaptive to a class of signals, 3) exact analysis and synthesis, i.e. allows for perfect signal reconstruction, 4) efficient training through resemblance with deep auto-encoder networks. The paper is presented well: it is clearly structured and well written. After a nice overview and introduction, the authors give a detailed derivation of their construction and show in a number of experiments the validity and versatility of their approach. The proposed approach and wavelet construction builds on previous work but makes a non-trivial contribution to the field of graph-based signal processing by deriving a general-purpose wavelet transform with a number of favourable properties.

Data stemming from networks, however, does not exhibit a regular inherent structure that can be effectively exploited by convolutions.

Although traffic prediction has been receiving considerable attention with a number of successes in the context of intelligent transportation systems, the prediction of traffic states over a complex transportation network that contains different road types has remained a challenge. This study proposes a multi-scale graph wavelet temporal convolution network (MSGWTCN) to predict the traffic states in complex transportation networks. Specifically, a multi-scale spatial block is designed to simultaneously capture the spatial information at different levels, and the gated temporal convolution network is employed to extract the temporal dependencies of the data. The model jointly learns to mount multiple levels of the spatial interactions by stacking graph wavelets with different scales. Two real-world datasets are used in this study to investigate the model performance, including a highway network in Seattle and a dense road network of Manhattan in New York City. Experiment results show that the proposed model outperforms other baseline models. Furthermore, different scales of graph wavelets are found to be effective in extracting local, intermediate and global information at the same time and thus enable the model to learn a complex transportation network topology with various types of road segments. By carefully customizing the scales of wavelets, the model is able to improve the prediction performance and better adapt to different network configurations.

Spectral graph convolution, an important tool of data filtering on graphs, relies on two essential decisions; selecting spectral bases for signal transformation and parameterizing the kernel for frequency analysis. While recent techniques mainly focus on standard Fourier transform and vector-valued spectral functions, they fall short in flexibility to describe specific signal distribution for each node, and expressivity of spectral function. In this paper, we present a novel wavelet-based graph convolution network, namely WaveGC, which integrates multi-resolution spectral bases and a matrix-valued filter kernel. Theoretically, we establish that WaveGC can effectively capture and decouple short-range and long-range information, providing superior filtering flexibility, surpassing existing graph convolutional networks and graph Transformers (GTs). To instantiate WaveGC, we introduce a novel technique for learning general graph wavelets by separately combining odd and even terms of Chebyshev polynomials. This approach strictly satisfies wavelet admissibility criteria. Our numerical experiments showcase the capabilities of the new network. By replacing the Transformer part in existing architectures with WaveGC, we consistently observe improvements in both short-range and long-range tasks. This underscores the effectiveness of the proposed model in handling different scenarios. Our code is available at https://github.com/liun-online/WaveGC.

Collaborative filtering is a popular approach in recommender systems, whose objective is to provide personalized item suggestions to potential users based on their purchase or browsing history. However, personalized recommendations require considerable amount of behavioral data on users, which is usually unavailable for new users, giving rise to the cold-start problem. To help alleviate this challenging problem, we introduce a spectral graph wavelet collaborative filtering framework for implicit feedback data, where users, items and their interactions are represented as a bipartite graph. Specifically, we first propose an adaptive transfer function by leveraging a power transform with the goal of stabilizing the variance of graph frequencies in the spectral domain. Then, we design a deep recommendation model for efficient learning of low-dimensional embeddings of users and items using spectral graph wavelets in an end-to-end fashion. In addition to capturing the graph's local and global structures, our approach yields localization of graph signals in both spatial and spectral domains, and hence not only learns discriminative representations of users and items, but also promotes the recommendation quality. The effectiveness of our proposed model is demonstrated through extensive experiments on real-world benchmark datasets, achieving better recommendation performance compared with strong baseline methods.

Spectral graph convolutional networks (SGCNs) have been attracting increasing attention in graph representation learning partly due to their interpretability through the prism of the established graph signal processing framework. However, existing SGCNs are limited in implementing graph convolutions with rigid transforms that could not adapt to signals residing on graphs and tasks at hand. In this paper, we propose a novel class of spectral graph convolutional networks that implement graph convolutions with adaptive graph wavelets. Specifically, the adaptive graph wavelets are learned with neural network-parameterized lifting structures, where structure-aware attention-based lifting operations are developed to jointly consider graph structures and node features. We propose to lift based on diffusion wavelets to alleviate the structural information loss induced by partitioning non-bipartite graphs. By design, the locality and sparsity of the resulting wavelet transform as well as the scalability of the lifting structure for large and varying-size graphs are guaranteed. We further derive a soft-thresholding filtering operation by learning sparse graph representations in terms of the learned wavelets, which improves the scalability and interpretablity, and yield a localized, efficient and scalable spectral graph convolution. To ensure that the learned graph representations are invariant to node permutations, a layer is employed at the input of the networks to reorder the nodes according to their local topology information. We evaluate the proposed networks in both node-level and graph-level representation learning tasks on benchmark citation and bioinformatics graph datasets. Extensive experiments demonstrate the superiority of the proposed networks over existing SGCNs in terms of accuracy, efficiency and scalability.

Breast cancer is one of the most common cancers in women worldwide, and early detection can significantly reduce the mortality rate of breast cancer. It is crucial to take multi-scale information of tissue structure into account in the detection of breast cancer. And thus, it is the key to design an accurate computer-aided detection (CAD) system to capture multi-scale contextual features in a cancerous tissue. In this work, we present a novel graph convolutional neural network for histopathological image classification of breast cancer. The new method, named multi-scale graph wavelet neural network (MS-GWNN), leverages the localization property of spectral graph wavelet to perform multi-scale analysis. By aggregating features at different scales, MS-GWNN can encode the multi-scale contextual interactions in the whole pathological slide. Experimental results on two public datasets demonstrate the superiority of the proposed method. Moreover, through ablation studies, we find that multi-scale analysis has a significant impact on the accuracy of cancer diagnosis.

Although spatiotemporal graph neural networks have achieved great empirical success in handling multiple correlated time series, they may be impractical in some real-world scenarios due to a lack of sufficient high-quality training data. Furthermore, spatiotemporal graph neural networks lack theoretical interpretation. To address these issues, we put forth a novel mathematically designed framework to analyze spatiotemporal data. Our proposed spatiotemporal graph scattering transform (ST-GST) extends traditional scattering transforms to the spatiotemporal domain. It performs iterative applications of spatiotemporal graph wavelets and nonlinear activation functions, which can be viewed as a forward pass of spatiotemporal graph convolutional networks without training. Since all the filter coefficients in ST-GST are mathematically designed, it is promising for the real-world scenarios with limited training data, and also allows for a theoretical analysis, which shows that the proposed ST-GST is stable to small perturbations of input signals and structures. Finally, our experiments show that i) ST-GST outperforms spatiotemporal graph convolutional networks by an increase of 35% in accuracy for MSR Action3D dataset; ii) it is better and computationally more efficient to design the transform based on separable spatiotemporal graphs than the joint ones; and iii) the nonlinearity in ST-GST is critical to empirical performance. Processing and learning from spatiotemporal data have received increasing attention recently. Examples include: i) skeleton-based human action recognition based on a sequence of human poses (Liu et al. (2019)), which is critical to human behavior understanding (Borges et al. (2013)), and ii) multi-agent trajectory prediction (Hu et al. (2020)), which is critical to robotics and autonomous driving (Shalev-Shwartz et al. (2016)). A common pattern across these applications is that data evolves in both spatial and temporal domains.

Variants of Graph Neural Networks (GNNs) for representation learning have been proposed recently and achieved fruitful results in various fields. Among them, graph attention networks (GATs) first employ a self-attention strategy to learn attention weights for each edge in the spatial domain. However, learning the attentions over edges only pays attention to the local information of graphs and greatly increases the number of parameters. In this paper, we first introduce attentions in the spectral domain of graphs. Accordingly, we present Spectral Graph Attention Network (SpGAT) that learn representations for different frequency components regarding weighted filters and graph wavelets bases. In this way, SpGAT can better capture global patterns of graphs in an efficient manner with much fewer learned parameters than that of GAT. We thoroughly evaluate the performance of SpGAT in the semi-supervised node classification task and verified the effectiveness of the learned attentions in the spectral domain.

Graph Domain Adaptation with Localized Graph Signal Representations Yusuf Yi git Pilavcı, Eylem Tu g ce G uneyi, Cemil Cengiz and Elif Vural Abstract In this paper we propose a domain adaptation algorithm designed for graph domains. Given a source graph with many labeled nodes and a target graph with few or no labeled nodes, we aim to estimate the target labels by making use of the similarity between the characteristics of the variation of the label functions on the two graphs. Our assumption about the source and the target domains is that the local behaviour of the label function, such as its spread and speed of variation on the graph, bears resemblance between the two graphs. We estimate the unknown target labels by solving an optimization problem where the label information is transferred from the source graph to the target graph based on the prior that the projections of the label functions onto localized graph bases be similar between the source and the target graphs. In order to efficiently capture the local variation of the label functions on the graphs, spectral graph wavelets are used as the graph bases. Experimentation on various data sets shows that the proposed method yields quite satisfactory classification accuracy compared to reference domain adaptation methods. Keywords: Domain adaptation, spectral graph theory, graph signal processing, spectral graph wavelets, graph Laplacian 1 Introduction A common assumption in machine learning is that the training and the test data are sampled from the same distribution. Domain adaptation methods aim to provide solutions to machine learning problems by dealing with this distribution discrepancy. In domain adaptation, a source domain and a target domain are considered where the label information is mostly available for the data samples in the source domain, and few or none of the class labels are known in the target domain. The purpose is then to improve the learning performance in the target domain by making use Y. Y. Pilavcı is with the GIPSA Lab at Universit e Grenoble Alpes, Grenoble. C. Cengiz is with the Dept. of Computer Science and Engineering at Ko c University, Istanbul. Most part of this work was performed while the authors were at METU. 1 arXiv:1911.02883v1 A variety of approaches have been proposed so far for the domain adaptation problem. Some methods are based on reweighing the samples for removing the sample selection bias [1, 2]. Another common solution is to align the source and the target domains through feature space mappings.