A Unified View on Graph Neural Networks as Graph Signal Denoising (2010.01777v2)

Abstract: Graph Neural Networks (GNNs) have risen to prominence in learning representations for graph structured data. A single GNN layer typically consists of a feature transformation and a feature aggregation operation. The former normally uses feed-forward networks to transform features, while the latter aggregates the transformed features over the graph. Numerous recent works have proposed GNN models with different designs in the aggregation operation. In this work, we establish mathematically that the aggregation processes in a group of representative GNN models including GCN, GAT, PPNP, and APPNP can be regarded as (approximately) solving a graph denoising problem with a smoothness assumption. Such a unified view across GNNs not only provides a new perspective to understand a variety of aggregation operations but also enables us to develop a unified graph neural network framework UGNN. To demonstrate its promising potential, we instantiate a novel GNN model, ADA-UGNN, derived from UGNN, to handle graphs with adaptive smoothness across nodes. Comprehensive experiments show the effectiveness of ADA-UGNN.

• The paper proposes a unified mathematical view connecting various GNN aggregation operations (GCN, GAT, PPNP, APPNP) to graph signal denoising problems.
• It shows that these different GNNs can be seen as solving graph denoising tasks with Laplacian regularization, often via iterative updates like gradient descent.
• Based on this view, the authors introduce the Ugnn framework and the Ada-Ugnn model, demonstrating enhanced performance, especially with adaptive local smoothing.

A Unified View on Graph Neural Networks as Graph Signal Denoising

Graph Neural Networks (GNNs) have become increasingly important for analyzing graph-structured data due to their ability to learn representations that support various tasks, such as node classification and graph classification. Despite the diversity in design choices among GNN models, particularly in their feature aggregation operations, a significant portion of their mathematical framework can be unified under the perspective of graph signal denoising.

This paper presents a mathematical foundation connecting the aggregation operations of several prominent GNN models, including Graph Convolutional Networks (GCN), Graph Attention Networks (GAT), Personalized Propagation of Neural Predictions (PPNP), and its approximation APPNP, to graph denoising problems with a smoothness assumption. The authors propose that these aggregation processes can be uniformly viewed as solutions to graph signal denoising tasks characterized by Laplacian regularization. This connection offers a consistent perspective on various aggregation techniques and provides a basis for developing a unified framework for GNN design, termed Ugnn.

Mathematical Framework and Unified Perspective

The paper illustrates that the aggregation operations in GCN, GAT, PPNP, and APPNP can be seen as graph signal denoising operations with different structures and assumptions:

• GCN: The aggregation operation in GCN can be viewed as solving a graph signal denoising problem using one-step gradient descent.
• GAT: Similarly, GAT adopts node-dependent smoothness factors, modeling attention scores as adaptive local smoothness enforced via one-step gradient descent.
• PPNP and APPNP: These models are presented as addressing the graph denoising problem either exactly (PPNP) or approximately (APPNP via iterative updates).

The key insight was to treat feature aggregation operations as solutions to graph signal denoising problems, where node features are smoothed across connected nodes with or without adaptive considerations for local smoothness.

Unified GNN Framework: Ugnn

Based on the denoising paradigm, the authors propose the Ugnn framework for GNN layer construction. This framework comprises:

1. Designing a graph regularization term based on specific application requirements;
2. Implementing feature transformations leveraging learned representations;
3. Aggregating transformed features across the graph by solving the graph signal denoising problem iteratively or recursively.

Instantiation: Ada-Ugnn Model

To demonstrate the potential of Ugnn, the paper introduces Ada-Ugnn, a novel GNN model that exhibits adaptive local smoothing capabilities. Ada-Ugnn aims to accommodate graphs with varying node smoothness properties, offering enhancements over existing methods without relying on uniform smoothness assumptions. Comprehensive experiments reveal that Ada-Ugnn outperforms baseline methods, especially in scenarios where node smoothness is heterogeneous or impacted by adversarial perturbations.

Implications and Future Directions

The unified view of GNNs through graph signal denoising opens avenues for novel architectural designs utilizing different regularization strategies suited to distinct application domains and graph properties. By acknowledging the importance of adaptive smoothness in real-world networks potentially subjected to noise or adversarial attacks, this research encourages further exploration into specialized aggregation operations tailored to specific datasets or desired smoothness characteristics.

Future research could explore enriching the Ugnn framework with more sophisticated denoising paradigms, extending its capabilities to other tasks beyond node classification, such as graph generation or dynamic graph modeling, thereby advancing theoretical and practical aspects of GNNs.