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A Unified View on Graph Neural Networks as Graph Signal Denoising
Yao Ma, Xiaorui Liu, Tong Zhao, Yozen Liu, Jiliang Tang, Neil Shah
TL;DR
The paper addresses whether differing GNN aggregation operations share a common mathematical basis. It unifies GCN, GAT, PPNP, and APPNP as exact or approximate graph-denoising procedures, then derives UGNN and Ada-UGNN; experiments report that Ada-UGNN handles varying local smoothness effectively.
Problem
Existing GNNs use varied aggregation designs, motivating a mathematical connection among their assumptions and a general framework for graphs with differing smoothness.
Method
The paper interprets representative GNN aggregation operations as graph signal denoising and uses regularizer design to construct UGNN and adaptive-smoothness Ada-UGNN.
Results
Ada-UGNN consistently outperforms GCN and GAT and outperforms or matches APPNP across datasets.
Takeaways & Limitations
The denoising perspective provides a unified way to understand existing GNN aggregations and design models for adaptive local smoothness.
Takeaways & Limitations
Ada-UGNN is not specifically designed to defend against adversarial attacks, and the paper does not claim it is the most suitable defense.
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
The paper asks whether diverse GNN aggregation operations share an intrinsic connection and establishes that representative models can be understood through graph signal denoising. This unified view motivates UGNN and its adaptive-smoothness instance Ada-UGNN.
- 1 INTRODUCTION: Different GNN models commonly share feature transformations but use distinct aggregation designs over graph-connected features.The paper identifies aggregation as the main source of architectural variation among representative GNNs.
- 1 INTRODUCTION: The paper mathematically connects GCN, GAT, PPNP, and APPNP aggregation with graph signal denoising under Laplacian smoothness regularization.The connections are exact for some models and approximate for others.
- 1 INTRODUCTION: This connection offers a uniform perspective for understanding representative aggregation operations and supports a general framework for designing new GNNs.The proposed framework is intended to accommodate different applications and graph properties.
- 1 INTRODUCTION: Ada-UGNN instantiates UGNN to handle graphs whose smoothness properties vary across nodes.Experiments are conducted to demonstrate the model’s effectiveness.
2 REPRESENTATIVE GRAPH NEURAL NETWORKS
Representative GNNs combine feature transformation with graph-based aggregation, but differ in how neighborhood information is propagated. GCN uses normalized adjacency, GAT uses learned attention, and PPNP/APPNP use personalized propagation.
- 2 REPRESENTATIVE GRAPH NEURAL NETWORKS: A GNN layer transforms input features X into X′ and aggregates those transformed features over graph G to produce updated node features H.Input and output feature matrices have node-by-feature dimensions R^N×d_in and R^N×d_out.
- 2.1 Graph Convolutional Networks: GCN uses a normalized adjacency matrix with self-loops for feature aggregation, and multiple layers can be stacked with nonlinear activations between them.The normalized adjacency is constructed from A + I and its corresponding degree matrix.
- 2.2 Graph Attention Networks: GAT aggregates self-inclusive neighborhood features using learnable attention scores that differentiate the importance of distinct neighboring nodes.The attention scores are normalized versions of learned pairwise scores, and GAT models can also be stacked.
- 2.3 Personalized Propagation of Neural Predictions: PPNP uses a Personalized PageRank matrix for aggregation and an MLP for feature transformation.The PPR matrix specifies the influence of node i on node j through a propagation parameter α.
- 2.3 Personalized Propagation of Neural Predictions: APPNP approximates PPNP by performing aggregation iteratively, with H^(0) = X′ and H^(K) converging to the exact PPNP solution as K increases.This iterative design avoids the costly matrix inverse used by PPNP.
3 GNNS AS GRAPH SIGNAL DENOISING
The paper formulates GNN aggregation as graph signal denoising and shows exact or approximate connections for PPNP, APPNP, GCN, and GAT. This perspective explains smoothness control and motivates node-adaptive aggregation.
- 3 GNNS AS GRAPH SIGNAL DENOISING: Graph signal denoising recovers a clean graph signal F from noisy signal S by balancing fidelity to S against Laplacian-regularized smoothness over G.The regularization measures differences between connected nodes and can also be interpreted as a sum of local smoothness terms.
- 3.1 Connection to PPNP and APPNP: PPNP exactly solves the denoising problem with normalized Laplacian L = I − A_tilde, using X′ as the noisy signal and c = α^-1.The result follows from the closed-form optimizer of the convex denoising objective.
- 3.1 Connection to PPNP and APPNP: A smaller APPNP or PPNP α corresponds to larger c and therefore enforces smoother output features over the graph.The denoising interpretation gives α a direct smoothness-control meaning.
- 3.2 Connection to GCN: GCN aggregation is one-step gradient descent for graph denoising, while stacked GCN layers solve successive denoising problems with transformed features as inputs.With the specified stepsize, the update becomes F ← A_tilde X′, matching GCN aggregation.
- 3.3 Connection to GAT: GAT is closely connected to approximate denoising with node-dependent smoothness, where learned attention scores correspond to normalized functions of node smoothness parameters.Multi-layer GAT can consequently be viewed as solving a series of graph denoising problems.
4 UGNN: A UNIFIED GNN FRAMEWORK VIA GRAPH SIGNAL DENOISING
UGNN unifies GNN aggregation operations as solutions to generalized graph signal denoising problems, with regularization terms encoding graph-dependent smoothness assumptions. It provides a design procedure for new aggregation operations, including Ada-UGNN for adaptive local smoothness.
- PPNP, APPNP, GCN, and GAT aggregation operations can be connected to graph signal denoising with generalized Laplacian regularization.PPNP, APPNP, and GCN enforce global smoothness, whereas GAT enforces local smoothness for each node.
- UGNN defines aggregation by selecting a regularization term r(C, F, G) that encodes graph-based priors on the clean features F.The parameter C may be scalar, vector, or matrix-valued, enabling global, node-wise, or edge-wise flexibility.
- Different regularization choices recover existing aggregation operations and can incorporate models such as PairNorm and DropEdge.These examples remain related to Laplacian regularization, while trend filtering illustrates a non-Laplacian option for piece-wise linear signals.
- UGNN designs GNN layers by specifying graph regularization, transforming features, and solving the generalized denoising problem for aggregation.The transformed features X′ serve as the denoising problem's input S.
- Ada-UGNN instantiates UGNN with a regularization term enforcing adaptive local smoothness across nodes.This model is introduced as an example of deriving a new GNN aggregation operation from the unified framework.
5 ADA-UGNN: ADAPTIVE LOCAL SMOOTHING WITH UGNN
Ada-UGNN addresses graphs whose nodes exhibit different neighborhood smoothness by learning node-specific smoothness factors within UGNN. Its aggregation is obtained through iterative gradient descent on the corresponding generalized denoising problem.
- Motivation: Global smoothing used by PPNP, APPNP, and GCN may not fit graphs where nodes have substantially different neighborhood smoothness.Real-world datasets show varying local label smoothness, with some nodes having noisier neighborhoods than others.
- Adaptive Local Smoothness: Ada-UGNN generalizes the regularization used by PPNP, APPNP, and GCN by assigning each node a smoothness factor C_i.C_i is treated as a positive scalar controlling node i's level of local smoothness.
- Feature Transformation: Ada-UGNN uses an MLP for feature transformation, producing X′ with output dimension equal to the number of classes for node classification.This follows the feature-transformation design adopted by PPNP and APPNP.
- Feature Aggregation: The Ada-UGNN aggregation is modeled by iterative gradient-descent updates applied for K iterations to solve the generalized denoising problem.The update uses adaptive step sizes b_i, with b_i calculated from neighboring smoothness factors.
- Learning Smoothness Factors: Because independently tuning N node-specific factors is impractical without prior knowledge, Ada-UGNN parameterizes C_i from node i's neighborhood information.The paper uses a transformation h_2 followed by h_1 and a sigmoid, with s controlling the upper bound of C_i.
- Feature Aggregation: After K iterations, the node-classification representation H^(K) is row-wise softmax normalized to produce each node's discrete class distribution.The initialization is H^(0)=X′.
6 EXPERIMENTAL EVALUATION
Experiments evaluate Ada-Ugnn across diverse graph datasets, local smoothness regimes, and adversarial perturbations. Ada-Ugnn generally outperforms or matches APPNP, with its strongest advantages when local smoothness varies across nodes.
- Datasets and Experimental Settings: Experiments use 10 datasets spanning citation, social, co-authorship, transportation, and co-purchase networks.Citation, social, and transportation results average 30 random seeds; co-authorship and co-purchase experiments use repeated splits and initializations.
- Datasets and Experimental Settings: The datasets exhibit substantial variation in local label smoothness, motivating adaptive smoothness modeling.Figure 1 highlights especially diverse smoothness distributions in BlogCatalog, Flickr, and Air-USA.
- Performance Comparison: Ada-Ugnn consistently outperforms GCN and GAT, while outperforming or matching APPNP across datasets.The authors attribute the advantage over APPNP to adaptive rather than fixed smoothness assumptions.
- Performance Comparison: Ada-Ugnn’s improvement over APPNP is strongest on datasets with diverse local smoothness and marginal when most nodes have perfect homophily.Strongly significant improvements over APPNP occur on some diverse or less-diverse datasets, whereas gains are marginal on Amazon-Photo, Coauthor-PH, and Citeseer.
- Performance Comparison: Ada-Ugnn improves accuracy especially for low-smoothness nodes while maintaining comparable performance for high-smoothness nodes.This pattern is reported across most datasets; when few nodes have low smoothness, overall performance can remain similar to APPNP.
7 RELATED WORKS
Prior work connects GNN aggregation with graph signal processing and denoising, while this paper develops a denoising-based unification across representative models.
- Spectral-based GNNs use graph convolutions grounded in spectral theory, and these convolutions are closely related to graph signal processing.
- Low-pass graph filters can denoise graph signals, and many algorithms discussed in the paper can be viewed as low-pass graph filters.
- Earlier studies connect GCN aggregation with Laplacian-regularized graph denoising, interpreting it as a first-order approximation to the optimal solution.
- A concurrent work identifies optimization commonalities among some GNNs, whereas this paper uses signal denoising to explain the framework and includes GAT through adaptive local smoothness.
8 CONCLUSION
The paper unifies several representative GNN aggregation operations as graph denoising problems and uses this perspective to construct UGNN and Ada-UGNN.
- GCN, PPNP, APPNP, and GAT aggregations can be mathematically interpreted as exact or approximate solutions to graph denoising problems.
- UGNN is a general framework for developing flexible GNN models by designing regularizers from the graph denoising perspective.
- Ada-UGNN instantiates UGNN with a regularizer for adaptive local smoothness across nodes and is evaluated as a new GNN model.