The nature of heterophilous graphs is significantly different from that of homophilous graphs, which causes difficulties in early graph neural network (GNN) models and suggests aggregations beyond the one-hop neighborhood. In this article, we develop a new way to implement multiscale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further design a graph framelet neural network model permutation equivariant graph framelet augmented network (PEGFAN) based on our constructed graph framelets. The experiments are conducted on a synthetic dataset and nine benchmark datasets to compare the performance with other state-of-the-art models. The result shows that our model can achieve the best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.

Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling. One prominent example is the shearlet system. Our objective in this paper is threefold: We first develop a digital shearlet theory which is rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implies that shearlet theory provides a unified treatment of both the continuum and digital realms. Second, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying MATLAB toolbox called ShearLab (www.ShearLab.org) is provided. And, third, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is exemplarily demonstrated for the digital shearlet transform.

Hypergraph neural networks (HNNs) have shown promise in handling tasks characterized by high-order correlations, achieving notable success across various applications. However, there has been limited focus on heterophilic hypergraph learning (HHL), in contrast to the increasing attention given to graph neural networks designed for graphs exhibiting heterophily. This paper aims to pave the way for HHL by addressing key gaps from multiple perspectives: measurement, dataset diversity, and baseline model development. First, we introduce metrics to quantify heterophily in hypergraphs, providing a numerical basis for assessing the homophily/heterophily ratio. Second, we develop diverse benchmark datasets across various real-world scenarios, facilitating comprehensive evaluations of existing HNNs and advancing research in HHL. Additionally, as a novel baseline model, we propose HyperUFG, a framelet-based HNN integrating both low-pass and high-pass filters. Extensive experiments conducted on synthetic and benchmark datasets highlight the challenges current HNNs face with heterophilic hypergraphs, while showcasing that HyperUFG performs competitively and often outperforms many existing models in such scenarios. Overall, our study underscores the urgent need for further exploration and development in this emerging field, with the potential to inspire and guide future research in HHL.