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Linear Reweighted Regularization Algorithms for Graph Matching Problem

Rongxuan Li

TL;DR

This work addresses graph matching, a challenging NP-hard problem, by relaxing permutation constraints to a convex set and introducing a linear reweighted regularization that preserves convexity. The core idea minimizes $f(X) = ||A X - X B||_F^2$ over a convex domain while adding a linear term $h_W(X)$ with iteratively updated weights, solved via a projected gradient method and accelerated projection onto the doubly stochastic set. A convergence theory shows that, under suitable conditions on the iterate proximity and regularization strength, the method converges to the graph matching solution; a practical variant updates weights and solves a sequence of convex subproblems, yielding sparse, high-quality solutions. The approach is demonstrated on synthetic networks and shapes, highlighting efficient sparsity-inducing behavior and potential for scalable graph matching in network alignment and computer vision tasks. Overall, the method provides a convexity-preserving, computationally efficient framework for sparse graph matching with broad applicability in pattern recognition and geometry processing.

Abstract

The graph matching problem is a significant special case of the Quadratic Assignment Problem, with extensive applications in pattern recognition, computer vision, protein alignments and related fields. As the problem is NP-hard, relaxation and regularization techniques are frequently employed to improve tractability. However, most existing regularization terms are nonconvex, posing optimization challenges. In this paper, we propose a linear reweighted regularizer framework for solving the relaxed graph matching problem, preserving the convexity of the formulation. By solving a sequence of relaxed problems with the linear reweighted regularization term, one can obtain a sparse solution that, under certain conditions, theoretically aligns with the original graph matching problem's solution. Furthermore, we present a practical version of the algorithm by incorporating the projected gradient method. The proposed framework is applied to synthetic instances, demonstrating promising numerical results.

Linear Reweighted Regularization Algorithms for Graph Matching Problem

TL;DR

This work addresses graph matching, a challenging NP-hard problem, by relaxing permutation constraints to a convex set and introducing a linear reweighted regularization that preserves convexity. The core idea minimizes over a convex domain while adding a linear term with iteratively updated weights, solved via a projected gradient method and accelerated projection onto the doubly stochastic set. A convergence theory shows that, under suitable conditions on the iterate proximity and regularization strength, the method converges to the graph matching solution; a practical variant updates weights and solves a sequence of convex subproblems, yielding sparse, high-quality solutions. The approach is demonstrated on synthetic networks and shapes, highlighting efficient sparsity-inducing behavior and potential for scalable graph matching in network alignment and computer vision tasks. Overall, the method provides a convexity-preserving, computationally efficient framework for sparse graph matching with broad applicability in pattern recognition and geometry processing.

Abstract

The graph matching problem is a significant special case of the Quadratic Assignment Problem, with extensive applications in pattern recognition, computer vision, protein alignments and related fields. As the problem is NP-hard, relaxation and regularization techniques are frequently employed to improve tractability. However, most existing regularization terms are nonconvex, posing optimization challenges. In this paper, we propose a linear reweighted regularizer framework for solving the relaxed graph matching problem, preserving the convexity of the formulation. By solving a sequence of relaxed problems with the linear reweighted regularization term, one can obtain a sparse solution that, under certain conditions, theoretically aligns with the original graph matching problem's solution. Furthermore, we present a practical version of the algorithm by incorporating the projected gradient method. The proposed framework is applied to synthetic instances, demonstrating promising numerical results.

Paper Structure

This paper contains 19 sections, 4 theorems, 36 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Regularization
  3. Relaxation
  4. Nonconvex Sparsity Regularization Terms
  5. Linear Reweighted Regularization Algorithmic Framework
  6. Linear Regularization Term
  7. Algorithm Description
  8. Convergence
  9. A Practical Reweighted Algorithm for Graph Matching Problem
  10. Projected Gradient Method
  11. Fast Dual Gradient Algorithm for Computing the Projection onto $\mathcal{D}_n$
  12. Application
  13. Network Alignment Review
  14. Graph in Social Network Alignment Problem
  15. Graph in Shape correspondence problem
  16. ...and 4 more sections

Key Result

Proposition 1

Let $f({\mathbf{X}})= \|{\mathbf{A}}{\mathbf{X}}-{\mathbf{X}}{\mathbf{B}}\|_{\mathsf{F}}^2$, then $f({\mathbf{X}})$ is convex on $\mathbb{R}^{n\times n}$.

Figures (3)

  • Figure 1: An illustration of the graph matching problem in social network alignment. The two graphs represent connections between individuals on different platforms. Facebook (left) and Twitter (right). The goal is to identify corresponding nodes (e.g. individuals) between the two graphs based on their structural connectivity.
  • Figure 1: Average and standard deviation results of Objective Error by linear reweighted, $L_{p=0.75}$ and $L_{p=0.5}$ regularization term, on solving 50 independently random generated instances of graph matching problem of dimension n = 50.
  • Figure 2: Average and standard deviation results of Residual by linear reweighted, $L_{p=0.75}$ and $L_{p=0.5}$ regularization term, on solving 50 independently random generated instances of graph matching problem of dimension n = 50.

Theorems & Definitions (9)

  • Proposition 1
  • Proof 1
  • Remark 1
  • Proposition 2
  • Lemma 1
  • Proof 2
  • Theorem 2
  • Proof 3
  • Remark 2