Various Graph Neural Networks (GNNs) have been successful in analyzing data
in non-Euclidean spaces, however, they have limitations such as oversmoothing,
i.e., information becomes excessively averaged as the number of hidden layers
increases. The issue stems from the intrinsic formulation of conventional graph
convolution where the nodal features are aggregated from a direct neighborhood
per layer across the entire nodes in the graph. As setting different number of
hidden layers per node is infeasible, recent works leverage a diffusion kernel
to redefine the graph structure and incorporate information from farther nodes.
Unfortunately, such approaches suffer from heavy diagonalization of a graph
Laplacian or learning a large transform matrix. In this regards, we propose a
diffusion learning framework, where the range of feature aggregation is
controlled by the scale of a diffusion kernel. For efficient computation, we
derive closed-form derivatives of approximations of the graph convolution with
respect to the scale, so that node-wise range can be adaptively learned. With a
downstream classifier, the entire framework is made trainable in an end-to-end
manner. Our model is tested on various standard datasets for node-wise
classification for the state-of-the-art performance, and it is also validated
on a real-world brain network data for graph classifications to demonstrate its
practicality for Alzheimer classification.

基于扩散核的扩散学习框架在图神经网络中控制了特征聚合范围，解决了节点级分类和过度平滑化的问题，并具备在阿尔茨海默病分类中实际应用的可行性。

学习在图上近似自适应核卷积

Learning to Approximate Adaptive Kernel Convolution on Graphs

Graph neural networks (GNNs) have become increasingly popular for
classification tasks on graph-structured data. Yet, the interplay between graph
topology and feature evolution in GNNs is not well understood. In this paper,
we focus on node-wise classification, illustrated with community detection on
stochastic block model graphs, and explore the feature evolution through the
lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep
classifiers (e.g. for image classification) beyond the zero training error
point, NC demonstrates a reduction in the deepest features' within-class
variability and an increased alignment of their class means to certain
symmetric structures. We start with an empirical study that shows that a
decrease in within-class variability is also prevalent in the node-wise
classification setting, however, not to the extent observed in the
instance-wise case. Then, we theoretically study this distinction.
Specifically, we show that even an "optimistic" mathematical model requires
that the graphs obey a strict structural condition in order to possess a
minimizer with exact collapse. Interestingly, this condition is viable also for
heterophilic graphs and relates to recent empirical studies on settings with
improved GNNs' generalization. Furthermore, by studying the gradient dynamics
of the theoretical model, we provide reasoning for the partial collapse
observed empirically. Finally, we present a study on the evolution of within-
and between-class feature variability across layers of a well-trained GNN and
contrast the behavior with spectral methods.

本文探讨了图神经网络中的特征演变与拓扑关系，并通过 “神经崩溃” 现象展示了节点分类中的该现象。作者进行了实证研究和理论分析，发现理论模型要求图满足一定严格的结构条件才能得到准确的收缩，并最终通过对图神经网络的研究比较了不同层之间的特征变化与光谱方法的差异。