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Symmetric Graph Convolutional Autoencoder for Unsupervised Graph Representation Learning
Jiwoong Park, Minsik Lee, Hyung Jin Chang, Kyuewang Lee, Jin Young Choi
TL;DR
Existing graph convolutional autoencoders have limitations in feature reconstruction and latent representation control. The paper addresses these issues with a symmetric graph convolutional autoencoder using stable Laplacian sharpening and an efficient subspace-clustering cost, with experiments validating the framework and reporting superior performance over various graph-based methods.
Problem
Existing graph convolutional autoencoders have limitations because some reconstruct affinity matrices, while MGAE's feature reconstruction uses no hidden layers and therefore cannot manipulate latent dimensionality.
Method
The paper builds a symmetric autoencoder using graph structure and node attributes in both processes, with Laplacian sharpening in the decoder and a subspace-clustering cost for image clustering.
Results
Ablation studies, comparisons with state-of-the-art methods, and visualizations validate the proposed components and report superior performance across the evaluated tasks.
Takeaways & Limitations
The framework extracts low-dimensional graph representations while supporting image clustering through jointly learned latent representations and latent affinity matrices.
Takeaways & Limitations
MGAE's feature reconstruction cannot manipulate latent dimensionality because it lacks hidden layers and performs a linear mapping.
Abstract
from arXiv · showhide
We propose a symmetric graph convolutional autoencoder which produces a low-dimensional latent representation from a graph. In contrast to the existing graph autoencoders with asymmetric decoder parts, the proposed autoencoder has a newly designed decoder which builds a completely symmetric autoencoder form. For the reconstruction of node features, the decoder is designed based on Laplacian sharpening as the counterpart of Laplacian smoothing of the encoder, which allows utilizing the graph structure in the whole processes of the proposed autoencoder architecture. In order to prevent the numerical instability of the network caused by the Laplacian sharpening introduction, we further propose a new numerically stable form of the Laplacian sharpening by incorporating the signed graphs. In addition, a new cost function which finds a latent representation and a latent affinity matrix simultaneously is devised to boost the performance of image clustering tasks. The experimental results on clustering, link prediction and visualization tasks strongly support that the proposed model is stable and outperforms various state-of-the-art algorithms.
1. Introduction
Graph convolutional autoencoders seek deep graph representations, but existing designs have limitations in feature reconstruction and latent-dimensional control. The paper proposes a symmetric architecture using graph structure throughout, a stable sharpening decoder, and an efficient subspace-clustering cost.
- Existing graph methods can be computationally inefficient or capture only shallow relationships between nodes.
- Existing VGAE-related models reconstruct affinity matrices, while MGAE reconstructs node features but cannot manipulate latent dimensionality because it uses a linear mapping.
- Its decoder uses Laplacian sharpening as the counterpart to encoder smoothing, with a numerically stable signed-graph formulation.
- The proposed autoencoder is completely symmetric and uses graph structure and node attributes throughout encoding and decoding.
- An efficient subspace-clustering cost jointly finds latent representations and latent affinity matrices for image clustering.
- Ablation studies and comparisons with state-of-the-art methods validate the components and report superior performance, alongside graph-clustering visualization.
2. Preliminaries
Graphs encode geometric relationships through nodes, edges, and affinity matrices, while spectral methods can be computationally costly. Spectral convolution addresses this by approximating graph filters with efficient polynomial operations and motivates GCN's Laplacian-smoothing interpretation.
- Basic graph notation: A graph G = (V, E, A) contains nodes, edges, and an affinity matrix A encoding pairwise node affinities.
- Spectral convolution: Spectral convolution applies a filter to graph signals in the Laplacian eigenbasis, but direct eigendecomposition is inappropriate for large graphs.
- Spectral convolution: Chebyshev-polynomial approximation avoids eigendecomposition and provides a localized, computationally efficient graph convolution.
- GCN formulation: GCN simplifies the approximation with K = 1 and λmax ≈ 2, then uses a renormalization trick based on self-loops for numerical stability.
- GCN formulation: GCN propagation has complexity O(mpc) for sparse graphs and jointly uses node features and graph structure.
- Laplacian smoothing: GCN can be interpreted as Laplacian smoothing: each node's representation becomes a weighted local average of itself and its neighbors.
3. The proposed method
GALA is a symmetric graph convolutional autoencoder whose encoder performs Laplacian smoothing and whose decoder performs Laplacian sharpening to reconstruct node features. A signed-graph formulation stabilizes sharpening, while an added subspace-clustering cost jointly learns latent representations and affinities for image clustering.
- 3.1. Laplacian sharpening: GALA is a symmetric graph convolutional autoencoder that extracts low-dimensional graph representations using Laplacian smoothing in the encoder and sharpening in the decoder.
- 3. The proposed method: The decoder reconstructs node feature matrices directly rather than affinity matrices, using graph structure and node attributes throughout encoding and decoding.
- 3.1. Laplacian sharpening: Laplacian sharpening moves each reconstructed node feature farther from its neighbors' centroid, complementing encoder smoothing and accelerating reconstruction.
- 3.2. Numerically stable Laplacian sharpening: The naive sharpening operator has spectral radius 3, so repeated application can make the network numerically unstable.
- 3.2. Numerically stable Laplacian sharpening: GALA uses a signed graph with ˆA = 2I_n − A and ˆD = 2I_n + D to obtain a sharpening operator with spectral radius 1.
- 3. The proposed method: Both encoder and decoder propagation functions have complexity O(mpc), making GALA computationally efficient when the affinity matrix is sparse.
- 3.3. Subspace clustering cost: For image clustering, GALA adds a subspace-clustering cost that jointly learns latent representations and a latent affinity matrix.
- 3.3. Subspace clustering cost: The subspace-clustering simplification replaces n×n matrix inversion with k×k inversion, reducing computational burden from O(n^3) to O(k^3).
4. Experiments
The experiments evaluate clustering and link prediction across network and image datasets using repeated runs and standard metrics.
- Datasets: Four network datasets and three image datasets are used for node clustering and link prediction tasks.Network datasets provide feature and affinity matrices, whereas image datasets provide feature matrices only.
- Metrics: Node clustering is evaluated with ACC, NMI, and ARI, reported as means over 50 executions.Higher values indicate more correct results.
- Reported results: The study reports dataset summaries and node-clustering experimental results in Tables 3 and 2, respectively.The supplied experimental-settings passages identify these tables but do not provide their numerical entries.
- Metrics: Link prediction uses mean AUC and AP scores with standard errors from 10 random initializations.The dataset partition follows the GAE protocol.
4.3. Comparing methods
The comparison covers 15 algorithms grouped by whether they use node features, network structure, both, or graph spectral convolution within autoencoders.
- Comparison groups: Fifteen algorithms are compared across four methodological groups.The groups distinguish feature-only, structure-only, combined, and combined spectral-convolution methods.
- Comparison groups: Kmeans represents feature-only clustering, while Spectral, Big-Clam, DeepWalk, GraEnc, and DNGR use network structure only.These methods span clustering, matrix factorization, neural-network, and graph-encoding approaches.
- Comparison groups: Circles, RTM, RMSC, and TADW use both features and network information without the listed graph spectral-convolution autoencoder formulation.Their approaches include social-circle discovery, relational topic modeling, multi-view clustering, and matrix factorization.
- Comparison groups: GAE, VGAE, MGAE, ARGA, and ARVGA combine graph spectral convolution with autoencoder-based representations.The group includes probabilistic, adversarial, and marginalization-based variants.
4.4. Node clustering results
Node-clustering results favor methods that jointly use features and network structure, with GALA outperforming the compared methods across reported datasets.
- Overall comparison: Methods using features and network structures simultaneously outperform methods using only one information source on every dataset.The results also identify spectral-convolution neural networks as particularly strong among combined methods.
- Overall comparison: GALA outperforms other methods in every reported experiment.The supplied passages describe this as applying to the node-clustering results.
- Cora: On Cora, GALA exceeds VGAE by 24.39% on ACC, 24.75% on NMI, and 27.68% on ARI.These are the reported comparisons against the first graph convolutional autoencoder framework.
- Cora: On Cora, GALA exceeds MGAE by 6.15% on ACC, 6.56% on NMI, and 8.68% on ARI.MGAE is identified in the passage as a state-of-the-art graph convolutional autoencoder algorithm.
- Pubmed: On Pubmed, GALA outperforms every baseline and state-of-the-art graph-convolution method despite Kmeans exceeding several graph-convolution methods on NMI and ARI.The passage does not provide the numerical Pubmed scores.
4.5. Image clustering results
Image-clustering experiments compare GALA with and without the subspace-clustering cost, showing that the added cost improves performance across all cases.
- Image clustering: GALA outperforms several baselines and state-of-the-art graph-convolution algorithms in most image-clustering cases.Results are reported for both reconstruction cost alone and the added subspace-clustering cost.
- Subspace clustering cost: The subspace-clustering cost improves image-clustering performance in every case.This cost jointly supports the latent representation and latent affinity matrix design described for image clustering.
- Subspace clustering cost: On YALE, the subspace-clustering cost significantly enhances image-clustering performance and achieves nearly perfect accuracy.The passage characterizes the YALE result qualitatively rather than giving an exact accuracy value.
4.6. Ablation studies
Image clustering ablations show that both the numerically stable decoder and the subspace clustering cost improve latent representations, with their combination providing the full proposed method.
- The stable decoder and subspace clustering cost each help latent representations reflect graph structure, and using both components boosts clustering performance.The reconstruction-only configuration is a subset of the full subspace clustering cost.
4.7. Link prediction results
The model outperforms compared methods on Citeseer link prediction and node clustering, extending its reported gains beyond clustering evaluations.
- Link prediction combines reconstruction with GAE's link prediction cost using the latent representation to reconstruct the affinity matrix.The reconstructed affinity matrix is defined as sigmoid(HH^T).
- GALA outperforms the compared methods on Citeseer link prediction and node clustering.
4.8. Visualization
Visualization results on YALE indicate that GALA produces embeddings with lower inter-cluster affinity and higher intra-cluster affinity than the compared methods.
- GALA embeds YALE nodes more effectively by minimizing inter-cluster affinity and maximizing intra-cluster affinity.
5. Conclusions
The paper presents a symmetric graph convolutional autoencoder that extracts low-dimensional graph representations using smoothing in the encoder and stable sharpening in the decoder. Experiments on network and image datasets support its validity and superior performance over graph-based clustering algorithms.
- The proposed framework extracts low-dimensional latent representations from graphs in irregular domains.
- The symmetric architecture pairs Laplacian smoothing in the encoder with Laplacian sharpening in the decoder.
- A signed-graph formulation gives Laplacian sharpening spectral radius one to prevent numerical instabilities.
- A subspace clustering cost term is added to the reconstruction cost for image clustering tasks.
- Experiments on network and image datasets show validity and superior performance over various graph-based clustering algorithms.