Learning Partial Graph Matching via Optimal Partial Transport

TL;DR

This work tackles partial graph matching, where only a subset of nodes across two graphs should be matched, by linking it to optimal partial transport (OPT).It introduces a POT-inspired objective that uses a weighted total variation divergence to promote valid, biased-aware partial assignments and proves an exact reduction to a linear sum assignment problem solvable by the Hungarian algorithm in $O(n^3)$ time.A deep graph matching architecture is proposed that learns cross-graph affinities, per-node matching biases, and a partial matching loss for end-to-end optimization.Empirical results on image keypoint benchmarks and protein interaction networks demonstrate strong accuracy and efficiency advantages over competitive baselines, with ablations highlighting the benefits of learned matching biases and the method's robustness under varying conditions.

Abstract

Partial graph matching extends traditional graph matching by allowing some nodes to remain unmatched, enabling applications in more complex scenarios. However, this flexibility introduces additional complexity, as both the subset of nodes to match and the optimal mapping must be determined. While recent studies have explored deep learning techniques for partial graph matching, a significant limitation remains: the absence of an optimization objective that fully captures the problem's intrinsic nature while enabling efficient solutions. In this paper, we propose a novel optimization framework for partial graph matching, inspired by optimal partial transport. Our approach formulates an objective that enables partial assignments while incorporating matching biases, using weighted total variation as the divergence function to guarantee optimal partial assignments. Our method can achieve efficient, exact solutions within cubic worst case time complexity. Our contributions are threefold: (i) we introduce a novel optimization objective that balances matched and unmatched nodes; (ii) we establish a connection between partial graph matching and linear sum assignment problem, enabling efficient solutions; (iii) we propose a deep graph matching architecture with a novel partial matching loss, providing an end-to-end solution. The empirical evaluations on standard graph matching benchmarks demonstrate the efficacy of the proposed approach.

Figures (3)

• Figure 1: An overview of our end-to-end learning architecture for partial graph matching. Given two input graphs $G_S$ and $G_T$, the neural network $NN_{\theta}$ generates the cross-graph node-to-node affinity matrix $A$. The affinity matrix $A$ is then used to learn the matching bias values $\alpha$, $\beta$, and also the cost matrix $C$. The partial matching loss is computed by taking into account the ground truth, $C$, $\alpha$, and $\beta$.
• Figure 2: (left) Performance (node correctness) of our model OPGM-rs against SIGMA and StableGM on PPI Network Matching with varying noise levels; (middle) Sensitivity analysis of our model OPGM-rs on the unbalancedness parameter $\rho$; (right) Sensitivity analysis of our model OPGM-rs on the regularization parameter $\lambda$. Performance in sensitivity analysis is measured by matching F1-score.
• Figure 3: Failure mode analysis across four different datasets.

Theorems & Definitions (21)

• Theorem 5.1: Infeasible Assignments
• Theorem 5.2: Existence of Optimal Solution
• Lemma 6.0: Equivalence
• Theorem 6.1: Optimal Solution
• Remark 6.2
• Proposition A.0: Equivalent cost of couplings
• proof
• Proposition A.0: Equivalence of minimizers
• proof
• Theorem A.1: Infeasible Assignments