Partition-wise Graph Filtering: A Unified Perspective Through the Lens of Graph Coarsening
Guoming Li, Jian Yang, Yifan Chen
TL;DR
This work addresses the limitations of purely graph-wise or node-wise filtering in graph neural networks by introducing Coarsening-guided Partition-wise Filtering (CPF). CPF performs filtering on node partitions derived from graph coarsening (structure-aware) and on clusters derived from feature embeddings (class-aware), unifying graph- and node-centric paradigms. The authors provide theoretical insights, including RSA-based guarantees and a unified embedding formulation Z = \sum_{k=0}^{K} diag(C^{+}\Theta_{:k+1}) T_{k}(\\boldsymbol{L}) X, and demonstrate strong empirical gains on both homophilic and heterophilic graphs, as well as real-world graph anomaly detection tasks. The approach highlights improved generalization and scalability, achieving state-of-the-art performance across diverse datasets while mitigating overfitting risks associated with fully node-wise filtering.
Abstract
Filtering-based graph neural networks (GNNs) constitute a distinct class of GNNs that employ graph filters to handle graph-structured data, achieving notable success in various graph-related tasks. Conventional methods adopt a graph-wise filtering paradigm, imposing a uniform filter across all nodes, yet recent findings suggest that this rigid paradigm struggles with heterophilic graphs. To overcome this, recent works have introduced node-wise filtering, which assigns distinct filters to individual nodes, offering enhanced adaptability. However, a fundamental gap remains: a comprehensive framework unifying these two strategies is still absent, limiting theoretical insights into the filtering paradigms. Moreover, through the lens of Contextual Stochastic Block Model, we reveal that a synthesis of graph-wise and node-wise filtering provides a sufficient solution for classification on graphs exhibiting both homophily and heterophily, suggesting the risk of excessive parameterization and potential overfitting with node-wise filtering. To address the limitations, this paper introduces Coarsening-guided Partition-wise Filtering (CPF). CPF innovates by performing filtering on node partitions. The method begins with structure-aware partition-wise filtering, which filters node partitions obtained via graph coarsening algorithms, and then performs feature-aware partition-wise filtering, refining node embeddings via filtering on clusters produced by $k$-means clustering over features. In-depth analysis is conducted for each phase of CPF, showing its superiority over other paradigms. Finally, benchmark node classification experiments, along with a real-world graph anomaly detection application, validate CPF's efficacy and practical utility.