Gromov-Wasserstein Learning for Graph Matching and Node Embedding (1901.06003v2)

Abstract: A novel Gromov-Wasserstein learning framework is proposed to jointly match (align) graphs and learn embedding vectors for the associated graph nodes. Using Gromov-Wasserstein discrepancy, we measure the dissimilarity between two graphs and find their correspondence, according to the learned optimal transport. The node embeddings associated with the two graphs are learned under the guidance of the optimal transport, the distance of which not only reflects the topological structure of each graph but also yields the correspondence across the graphs. These two learning steps are mutually-beneficial, and are unified here by minimizing the Gromov-Wasserstein discrepancy with structural regularizers. This framework leads to an optimization problem that is solved by a proximal point method. We apply the proposed method to matching problems in real-world networks, and demonstrate its superior performance compared to alternative approaches.

• The paper presents a unified framework that concurrently optimizes graph matching and node embedding by minimizing the Gromov-Wasserstein discrepancy.
• An innovative proximal point method iteratively updates optimal transport to align both graph topology and embedding spaces.
• Experimental results on network alignment tasks showcase the framework’s robust performance in handling noisy and densely connected graphs.

Overview of Gromov-Wasserstein Learning Framework for Joint Graph Matching and Node Embedding

The paper presents a Gromov-Wasserstein (GW) learning framework devised to concurrently address graph matching and node embedding learning. By capitalizing on the Gromov-Wasserstein discrepancy, the paper aims to assess dissimilarities between graphs and guide the matching process through the calculated optimal transport. The framework departs from conventional methodologies by treating graph matching and node embedding not as separate tasks but as interdependent processes, thereby enhancing accuracy and robustness in both domains.

Key to the paper's approach is using the GW discrepancy, which aids in aligning the topologies of different graphs by comparing their distance matrices in a relational deviation manner. Optimal transport, learned through the framework, not only bridges these topologies but also aligns node embeddings across different graphs, maintaining consistency with the optimal transport's guidance, thus enabling seamless correspondence determination and aiding the embedding process.

Technical Contributions

1. Unified Framework:
   • The proposed framework ingeniously integrates graph matching and node embeddings into a singular optimization challenge, orchestrated by minimizing the GW discrepancy with additional structural regularizers. Harmonizing these two tasks within the same framework promotes mutual reinforcement, fostering greater accuracy.
2. Algorithmic Innovation:
   • A proximal point method facilitates solving the optimization problem. The solution dynamically updates optimal transport based on node embeddings and graph topology iteratively, balancing both tasks efficiently.
3. Embeddings Regularization:
   • The framework introduces a regularization mechanism that leverages both the GW discrepancy and Wasserstein discrepancy to position node embeddings on the same manifold, circumventing explicit transformations or pre-defined constraints, thereby reducing the risk of model misspecification.

Experimental Validation

The framework's efficacy is validated in various real-world network matching scenarios, including network alignment tasks stemming from biological networks and communication networks. The experimental results demonstrate superior performance compared to alternative methodologies, offering improved robustness particularly in scenarios where graphs are noisy or consist of dense links.

Implications and Future Scope

The framework's fusion of node embedding learning and graph matching into a coherent strategy is notably beneficial:

• Pragmatically, it addresses the problematics of noisy graphs, a common ordeal in existing graph alignment strategies.
• Theoretically, it opens the floor for future exploration into multi-graph matching and further extensions of the GW discrepancy-based learning algorithms.

In conclusion, this paper not only introduces a robust framework for graph matching and node embedding but also sets foundational grounds for expanding GW-based learning methods in broader computational applications such as recommenders and beyond. The advancements, measured by the framework's concrete performance improvements, reflect significant potential for further academic inquiry and practical implementation in AI-driven graph analysis tasks.