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A quadratic penalty method for hypergraph matching

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Abstract

Hypergraph matching is a fundamental problem in computer vision. Mathematically, it maximizes a polynomial objective function, subject to assignment constraints. In this paper, we reformulate the hypergraph matching problem as a sparse constrained optimization problem. By dropping the sparse constraint, we show that the resulting relaxation problem can recover the global minimizer of the original problem. This property heavily depends on the special structures of hypergraph matching. The critical step in solving the original problem is to identify the location of nonzero entries (referred to as the support set) in a global minimizer. Inspired by such observation, we apply the quadratic penalty method to solve the relaxation problem. Under reasonable assumptions, we show that the support set of the global minimizer in a hypergraph matching problem can be correctly identified when the number of iterations is sufficiently large. A projected gradient method is applied as a subsolver to solve the quadratic penalty subproblem. Numerical results demonstrate that the exact recovery of the support set indeed happens, and the proposed algorithm is efficient in terms of both accuracy and CPU time.

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Notes

  1. \(\hbox {mean}(\Vert f_{l_1j_1k_1}-f_{l_2j_2k_2}\Vert )=\frac{\sum _{\mathcal {B}_{l_1l_2j_1j_2k_1k_2}>0}\Vert f_{l_1j_1k_1}-f_{l_2j_2k_2}\Vert }{\hbox {number of } B_{l_1l_2j_1j_2k_1k_2}>0}\).

  2. Downloaded from http://vasc.ri.cmu.edu/idb/html/motion/house/.

  3. Downloaded from http://www.umiacs.umd.edu/~zhengyf/PointMatching.htm.

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Acknowledgements

The authors would like to thank Dr. Yafeng Liu from Academy of Mathematics and Systems Science, Dr. Bo Jiang from Nanjing Normal University, and Dr. Lili Pan from Beijing Jiaotong University for discussions and insightful comments on this paper. We are also grateful to two anonymous reviewers for their valuable comments, which further improved the quality of this paper.

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Authors and Affiliations

  1. Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, China

    Chunfeng Cui & Hong Yan

  2. School of Mathematics and Statistics, Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, China

    Qingna Li

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China

    Liqun Qi

Authors
  1. Chunfeng Cui
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  2. Qingna Li
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  3. Liqun Qi
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  4. Hong Yan
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Corresponding author

Correspondence to Qingna Li.

Additional information

Chunfeng Cui was supported by the Research Grants Council (RGC) of Hong Kong (Project C1007-15G). Qingna Li was supported by NSFC 11671036. Liqun Qi was supported by the Research Grants Council (RGC) of Hong Kong (Project C1007-15G). Hong Yan was supported by the Research Grants Council (RGC) of Hong Kong (Project C1007-15G).

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Cui, C., Li, Q., Qi, L. et al. A quadratic penalty method for hypergraph matching. J Glob Optim 70, 237–259 (2018). https://doi.org/10.1007/s10898-017-0583-0

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