Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

CSGO: Constrained-Softassign Gradient Optimization For Large Graph Matching

Binrui Shen, Qiang Niu, Shengxin Zhu

TL;DR

This paper tackles large-scale graph matching by recasting the problem as a constrained-gradient optimization of the Koopmans–Beckmann QAP and unifying various classical methods under a single framework. It introduces Constrained-Softassign Gradient Optimization (CSGO), built on a scalable softassign constraining operator, an adaptive step-size rule, and a warm-start strategy to enable fast and robust matching on graphs with many nodes. Key contributions include a normalization/entropy-based softassign that is invariant to problem scale, a practical adaptive α that ensures monotone improvement, and a computationally efficient warm-start that dramatically reduces first-iteration cost. Together, these innovations yield substantial speedups (often an order of magnitude) and improved accuracy, particularly for attributed graphs, making large graph matching viable in real-world applications such as image matching and network alignment.

Abstract

Graph matching aims to find correspondences between two graphs. This paper integrates several well-known graph matching algorithms into a framework: the constrained gradient method. The primary difference among these algorithms lies in tuning a step size parameter and constraining operators. By leveraging these insights, we propose an adaptive step size parameter to guarantee the underlying algorithms' convergence, simultaneously enhancing their efficiency and robustness. For the constraining operator, we introduce a scalable softassign for large graph matching problems. Compared to the original softassign, our approach offers increased speed, improved robustness, and reduced risk of overflow. The advanced constraining operator enables a CSGO for large graph matching, which outperforms state-of-the-art methods in experiments. Notably, in attributed graph matching tasks, CSGO achieves an over 10X increase in speed compared to current constrained gradient algorithms.

CSGO: Constrained-Softassign Gradient Optimization For Large Graph Matching

TL;DR

This paper tackles large-scale graph matching by recasting the problem as a constrained-gradient optimization of the Koopmans–Beckmann QAP and unifying various classical methods under a single framework. It introduces Constrained-Softassign Gradient Optimization (CSGO), built on a scalable softassign constraining operator, an adaptive step-size rule, and a warm-start strategy to enable fast and robust matching on graphs with many nodes. Key contributions include a normalization/entropy-based softassign that is invariant to problem scale, a practical adaptive α that ensures monotone improvement, and a computationally efficient warm-start that dramatically reduces first-iteration cost. Together, these innovations yield substantial speedups (often an order of magnitude) and improved accuracy, particularly for attributed graphs, making large graph matching viable in real-world applications such as image matching and network alignment.

Abstract

Graph matching aims to find correspondences between two graphs. This paper integrates several well-known graph matching algorithms into a framework: the constrained gradient method. The primary difference among these algorithms lies in tuning a step size parameter and constraining operators. By leveraging these insights, we propose an adaptive step size parameter to guarantee the underlying algorithms' convergence, simultaneously enhancing their efficiency and robustness. For the constraining operator, we introduce a scalable softassign for large graph matching problems. Compared to the original softassign, our approach offers increased speed, improved robustness, and reduced risk of overflow. The advanced constraining operator enables a CSGO for large graph matching, which outperforms state-of-the-art methods in experiments. Notably, in attributed graph matching tasks, CSGO achieves an over 10X increase in speed compared to current constrained gradient algorithms.
Paper Structure (30 sections, 6 theorems, 48 equations, 9 figures, 9 tables, 3 algorithms)

This paper contains 30 sections, 6 theorems, 48 equations, 9 figures, 9 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Attributed graph and matching matrix
  4. Graph matching problem and relaxation
  5. Constrained gradient method
  6. Scalable softassign method
  7. Softassign and vectorized update
  8. Robustness through parameter tuning
  9. Stable strategy
  10. Algorithm of scalable softassign
  11. Analysis of step-size parameter
  12. Adaptive optimal step-size parameter
  13. Optimal step-size parameter for doubly stochastic constraining operators
  14. Constrained-Softassign Gradient Optimization Algorithm
  15. Warm-start strategy
  16. ...and 15 more sections

Key Result

Proposition 1

Let $X \in \mathbb{R}^{n \times n}$, $\beta \in \mathbb{R}$, and $Y=\mathcal{P}^{\beta}_{S}(X)$. Then

Figures (9)

  • Figure 2: The relation between $\alpha$ and objective score. The red point represents the score with optimal $\alpha$.
  • Figure 3: The plots of differences between the optimal solution and the updated matrix with respect to a number of iterations. Here, $P$ is the matrix computed at the current iteration; $P_{opt}$ is the optimal solution computed by the simplex method.
  • Figure 4: Graphs from real-world images.
  • Figure 5: Matching time of constrained gradient algorithms in attributed graph matching experiments.
  • Figure 6: Matching error of constrained gradient algorithms in attributed graph matching experiments.
  • ...and 4 more figures

Theorems & Definitions (15)

  • Proposition 1
  • Proof
  • Proposition 2
  • Proof
  • Proposition 3
  • Proof
  • Proof
  • Proof
  • Proof
  • Remark 1
  • ...and 5 more