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Robust Graph Matching Using An Unbalanced Hierarchical Optimal Transport Framework

Haoran Cheng, Dixin Luo, Hongteng Xu

TL;DR

This study proposes a novel and robust graph matching method based on an unbalanced hierarchical optimal transport (UHOT) framework, which, to the knowledge, makes the first attempt to exploit cross-modal alignment in graph matching.

Abstract

Graph matching is one of the most significant graph analytic tasks, which aims to find the node correspondence across different graphs. Most existing graph matching approaches mainly rely on topological information, whose performances are often sub-optimal and sensitive to data noise because of not fully leveraging the multi-modal information hidden in graphs, such as node attributes, subgraph structures, etc. In this study, we propose a novel and robust graph matching method based on an unbalanced hierarchical optimal transport (UHOT) framework, which, to our knowledge, makes the first attempt to exploit cross-modal alignment in graph matching. In principle, applying multi-layer message passing, we represent each graph as layer-wise node embeddings corresponding to different modalities. Given two graphs, we align their node embeddings within the same modality and across different modalities, respectively. Then, we infer the node correspondence by the weighted average of all the alignment results. This method is implemented as computing the UHOT distance between the two graphs -- each alignment is achieved by a node-level optimal transport plan between two sets of node embeddings, and the weights of all alignment results correspond to an unbalanced modality-level optimal transport plan. Experiments on various graph matching tasks demonstrate the superiority and robustness of our method compared to state-of-the-art approaches. Our implementation is available at https://github.com/Dixin-Lab/UHOT-GM.

Robust Graph Matching Using An Unbalanced Hierarchical Optimal Transport Framework

TL;DR

This study proposes a novel and robust graph matching method based on an unbalanced hierarchical optimal transport (UHOT) framework, which, to the knowledge, makes the first attempt to exploit cross-modal alignment in graph matching.

Abstract

Graph matching is one of the most significant graph analytic tasks, which aims to find the node correspondence across different graphs. Most existing graph matching approaches mainly rely on topological information, whose performances are often sub-optimal and sensitive to data noise because of not fully leveraging the multi-modal information hidden in graphs, such as node attributes, subgraph structures, etc. In this study, we propose a novel and robust graph matching method based on an unbalanced hierarchical optimal transport (UHOT) framework, which, to our knowledge, makes the first attempt to exploit cross-modal alignment in graph matching. In principle, applying multi-layer message passing, we represent each graph as layer-wise node embeddings corresponding to different modalities. Given two graphs, we align their node embeddings within the same modality and across different modalities, respectively. Then, we infer the node correspondence by the weighted average of all the alignment results. This method is implemented as computing the UHOT distance between the two graphs -- each alignment is achieved by a node-level optimal transport plan between two sets of node embeddings, and the weights of all alignment results correspond to an unbalanced modality-level optimal transport plan. Experiments on various graph matching tasks demonstrate the superiority and robustness of our method compared to state-of-the-art approaches. Our implementation is available at https://github.com/Dixin-Lab/UHOT-GM.
Paper Structure (28 sections, 1 theorem, 22 equations, 6 figures, 3 tables, 4 algorithms)

This paper contains 28 sections, 1 theorem, 22 equations, 6 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Proposed Method
  4. Preliminaries and Motivation
  5. Proposed UHOT Framework
  6. Multi-modal Information Extraction
  7. Node-level Optimal Transports across Different Modalities
  8. A Modality-level Unbalanced Optimal Transport
  9. Robust Graph Matching by Minimizing HOT
  10. Optimization Algorithm
  11. Advantages Compared to Existing Methods
  12. Compared to single-modal graph matching methods
  13. Compared to multi-modal graph matching methods
  14. Experiments
  15. Experimental Setup
  16. ...and 13 more sections

Key Result

Proposition 1

For simplifying notations, we define $\bm{D}_s^{all}:=\sum_{p=1}^M\bm{D}_s^{(p)}$ and $\bm{D}_t^{all}:=\sum_{q=1}^M\bm{D}_t^{(q)}$, respectively. When setting $\beta=1$ (using GW distance as the grounding cost), we have where $C$ is nonnegative and defined as

Figures (6)

  • Figure 1: The scheme of our method. Given two graphs, we extract their multi-modal information by multi-layer message passing. We align the node embeddings of the two graphs within the same modality and across different modalities, respectively, by solving a series of node-level OT problems. We fuse the alignment results by solving a modality-level UOT problem and infer node correspondence accordingly.
  • Figure 2: An illustration of our message passing-based multi-modal information extraction.
  • Figure 3: The convergence curve on PPI.
  • Figure 4: Comparisons on robustness and efficiency.
  • Figure 5: Testing on the robustness to hyperparameters.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof