[Paper Review] Gromov-Wasserstein Learning for Graph Matching and Node Embedding

Hongteng Xu, Dixin Luo+2 more|arXiv (Cornell University)|Jan 17, 2019

Advanced Graph Neural NetworksComputer Science51 references88 citations

TL;DR

The paper introduces Gromov-Wasserstein Learning to jointly match graphs and learn cross-graph node embeddings by minimizing a regularized GW discrepancy, solved via a proximal point method.

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.

Motivation & Objective

• Motivate robust graph matching when graphs are noisy or incomplete.
• Jointly learn node embeddings and a cross-graph transport plan guiding alignment.
• Unify graph matching and embedding learning under a Gromov-Wasserstein discrepancy with structural regularizers.

Proposed method

• Define GW discrepancy for two graphs with distance matrices derived from data and embeddings.
• Construct C_s and C_t as mixtures of data-driven and embedding-based distances (C_k(X_k) = (1-α)C_k + αK(X_k,X_k)).
• Incorporate Wasserstein discrepancy between embeddings across graphs via K(X_s,X_t).
• Regularize embeddings with a prior consistency loss R(X_s,X_t).
• Solve the nonconvex problem by alternating optimization over T (optimal transport) and X (embeddings) using a proximal point method with KL-divergence ( Sinkhorn-equivalent step).
• Initialization schedules set α to grow across outer iterations to stabilize joint learning.

Experimental results

Research questions

• RQ1 Can GW discrepancy be used to jointly align two graphs and learn cross-graph node embeddings?
• RQ2 Does incorporating embedding-based distances and regularizers improve robustness to noisy/partial graphs compared to data-only GW matching?
• RQ3 What is the effectiveness and stability of the proposed proximal point optimization for learning transport and embeddings?
• RQ4 How does the method scale and perform relative to state-of-the-art graph matching approaches on synthetic and real networks?

Key findings

• The method achieves near 100% node correctness on identical source/target graphs and GW discrepancy approaching zero in synthetic tests.
• Embedding-aware GWL (GWL-C, GWL-R) outperforms data-only GW matching (GWD) especially under higher noise levels.
• On the MC3 communication network dataset, GWL variants outperform a range of competitive graph matching methods on both sparse and dense graphs.
• Learned embeddings from GWL align on a common manifold across graphs, with overlapped embeddings indicating matched pairs.
• The approach offers favorable computational complexity and robustness, with parallelizable components on GPUs.