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CLAP: Concave Linear APproximation for Quadratic Graph Matching

Yongqing Liang, Huijun Han, Xin Li

TL;DR

This work introduces a novel linear model and solver designed to accelerate the computation of graph matching, employing a positive semi-definite matrix approximation to establish the structural attribute constraint.

Abstract

Solving point-wise feature correspondence in visual data is a fundamental problem in computer vision. A powerful model that addresses this challenge is to formulate it as graph matching, which entails solving a Quadratic Assignment Problem (QAP) with node-wise and edge-wise constraints. However, solving such a QAP can be both expensive and difficult due to numerous local extreme points. In this work, we introduce a novel linear model and solver designed to accelerate the computation of graph matching. Specifically, we employ a positive semi-definite matrix approximation to establish the structural attribute constraint.We then transform the original QAP into a linear model that is concave for maximization. This model can subsequently be solved using the Sinkhorn optimal transport algorithm, known for its enhanced efficiency and numerical stability compared to existing approaches. Experimental results on the widely used benchmark PascalVOC showcase that our algorithm achieves state-of-the-art performance with significantly improved efficiency. Source code: https://github.com/xmlyqing00/clap

CLAP: Concave Linear APproximation for Quadratic Graph Matching

TL;DR

This work introduces a novel linear model and solver designed to accelerate the computation of graph matching, employing a positive semi-definite matrix approximation to establish the structural attribute constraint.

Abstract

Solving point-wise feature correspondence in visual data is a fundamental problem in computer vision. A powerful model that addresses this challenge is to formulate it as graph matching, which entails solving a Quadratic Assignment Problem (QAP) with node-wise and edge-wise constraints. However, solving such a QAP can be both expensive and difficult due to numerous local extreme points. In this work, we introduce a novel linear model and solver designed to accelerate the computation of graph matching. Specifically, we employ a positive semi-definite matrix approximation to establish the structural attribute constraint.We then transform the original QAP into a linear model that is concave for maximization. This model can subsequently be solved using the Sinkhorn optimal transport algorithm, known for its enhanced efficiency and numerical stability compared to existing approaches. Experimental results on the widely used benchmark PascalVOC showcase that our algorithm achieves state-of-the-art performance with significantly improved efficiency. Source code: https://github.com/xmlyqing00/clap

Paper Structure

This paper contains 23 sections, 14 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Unary Matching
  4. Pairwise Matching
  5. Algorithm
  6. Problem Definition
  7. Our Model
  8. Symmetric Edge Matrices.
  9. Linear Matching Model.
  10. Graph Attribute Construction
  11. Node Similarity.
  12. Edge Similarity.
  13. CLAP Solver
  14. Experiments
  15. Evaluation Metrics.
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Graph matching results by unary matching and pairwise matching. The image undergoes an affine transformation. Green/red lines indicate correct/wrong matchings.
  • Figure 2: Qualitative comparisons on synthetic transformed images, where green/red lines indicate correct/wrong matchings, respectively. 'Adj', 'Len', and 'Learning' represent the three types of edge attributes adopted in qc-DGM and our models, respectively.