Implicit graph neural networks (GNNs) have emerged as a potential approach to enable GNNs to capture long-range dependencies effectively. However, poorly designed implicit GNN layers can experience over-smoothing or may have limited adaptability to learn data geometry, potentially hindering their performance in graph learning problems. To address these issues, we introduce a geometric framework to design implicit graph diffusion layers based on a parameterized graph Laplacian operator. Our framework allows learning the geometry of vertex and edge spaces, as well as the graph gradient operator from data. We further show how implicit GNN layers can be viewed as the fixed-point solution of a Dirichlet energy minimization problem and give conditions under which it may suffer from over-smoothing. To overcome the over-smoothing problem, we design our implicit graph diffusion layer as the solution of a Dirichlet energy minimization problem with constraints on vertex features, enabling it to trade off smoothing with the preservation of node feature information. With an appropriate hyperparameter set to be larger than the largest eigenvalue of the parameterized graph Laplacian, our framework guarantees a unique equilibrium and quick convergence. Our models demonstrate better performance than leading implicit and explicit GNNs on benchmark datasets for node and graph classification tasks, with substantial accuracy improvements observed for some datasets.

翻译：隐式图神经网络已成为使GNN有效捕获长程依赖关系的潜在方法。然而，设计不当的隐式GNN层可能出现过度平滑问题，或对学习数据几何结构的适应性有限，这可能阻碍其在图学习任务中的性能。针对这些问题，我们引入了一个基于参数化图拉普拉斯算子的几何框架来设计隐式图扩散层。该框架支持从数据中学习顶点和边空间的几何结构以及图梯度算子。我们进一步证明了隐式GNN层可视为狄利克雷能量最小化问题的不动点解，并给出了其可能遭受过度平滑问题的条件。为克服过度平滑问题，我们将隐式图扩散层设计为带顶点特征约束的狄利克雷能量最小化问题的解，使其能够在平滑处理与节点特征信息保持之间实现权衡。通过设置大于参数化图拉普拉斯算子最大特征值的超参数，该框架可保证唯一均衡解和快速收敛。在节点分类和图分类任务的基准数据集上，我们的模型性能优于主流隐式和显式GNN，部分数据集的准确率提升尤为显著。