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NGM

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NGM

Our implementation of the following paper:

  • Runzhong Wang, Junchi Yan, Xiaokang Yang. "Neural Graph Matching Network: Learning Lawler's Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching." TPAMI 2021. [paper], [project page]

NGM proposes a learnable solver to the Lawler's Quadratic Assignment Problem (Lawler's QAP), which is the most general form of graph matching. The learning of Lawler's QAP is achieved by firstly transforming the Lawler's QAP to the so-called association graph, and solving Lawler's QAP can be equivalently transformed into a vertex classification problem on the association graph. Finally, Graph Neural Networks are adopted to solve vertex classification problem as a common technique.

NGM has the following components:

  • (If images as the input) VGG16 CNN to extract image features
  • (If images as the input) Delaunay triangulation to build graphs
  • (If images as the input) Building affinity matrix efficiently via Factorized Graph Matching (FGM)
  • Solving the resulting Quadratic Assignment Problem (QAP) by the proposed Graph Neural Network solver
  • Supervised learning based on cross-entropy (also known as "permutation loss")

We also show that such a general scheme can be extended to hyper-graph matching (by building a so-called association hypergraph) and multi-graph matching (by adding a spectral multi-matching head).

The first version of NGM/NHGM/NMGM is developed in 2019 with VGG16 CNN in line with GMN. We further develop an improved version NGM-v2/NHGM-v2/NMGM-v2 in 2021 by simply modifying the VGG16 to the improved backbone developed by BBGM.

To summarize, here we include the following models:

  • the original models developed in 2019:
    • NGM: two-graph matching/Lawler's QAP solver with VGG16
    • NHGM: hyper-graph matching solver with VGG16
    • NMGM: multi-graph matching solver with VGG16
  • the improved v2 models developed in 2021:
    • NGM-v2: two-graph matching/Lawler's QAP solver with VGG16+SplineConv
    • NHGM-v2: hyper-graph matching solver with VGG16+SplineConv
    • NMGM-v2: multi-graph matching solver with VGG16+SplineConv

Benchmark Results

PascalVOC - 2GM

  • NGM

    experiment config: experiments/vgg16_ngm_voc.yaml

    pretrained model: google drive

  • NHGM

    experiment config: experiments/vgg16_nhgm_voc.yaml

    pretrained model: google drive

  • NGM-v2

    experiment config: experiments/vgg16_ngmv2_voc.yaml

    pretrained model: google drive

  • NHGM-v2

    experiment config: experiments/vgg16_nhgmv2_voc.yaml

    pretrained model: google drive

Since NMGM does not support partial matching, we do not evaluate NMGM on Pascal VOC.

model year aero bike bird boat bottle bus car cat chair cow table dog horse mbkie person plant sheep sofa train tv mean
NGM 2019 0.5010 0.6350 0.5790 0.5340 0.7980 0.7710 0.7360 0.6820 0.4110 0.6640 0.4080 0.6030 0.6190 0.6350 0.4560 0.7710 0.6930 0.6550 0.7920 0.8820 0.6413
NHGM 2019 0.5240 0.6220 0.5830 0.5570 0.7870 0.7770 0.7440 0.7070 0.4200 0.6460 0.5380 0.6100 0.6190 0.6080 0.4680 0.7910 0.6680 0.5510 0.8090 0.8870 0.6458
NGM-v2 2021 0.6184 0.7118 0.7762 0.7875 0.8733 0.9363 0.8770 0.7977 0.5535 0.7781 0.8952 0.7880 0.8011 0.7923 0.6258 0.9771 0.7769 0.7574 0.9665 0.9323 0.8011
NHGM-v2 2021 0.5995 0.7154 0.7724 0.7902 0.8773 0.9457 0.8903 0.8181 0.5995 0.8129 0.8695 0.7811 0.7645 0.7750 0.6440 0.9872 0.7778 0.7538 0.9787 0.9280 0.8040

Willow Object Class - 2GM & MGM

  • NGM

    experiment config: experiments/vgg16_ngm_willow.yaml

    pretrained model: google drive

  • NHGM

    experiment config: experiments/vgg16_nhgm_willow.yaml

    pretrained model: google drive

  • NMGM

    experiment config: experiments/vgg16_nmgm_willow.yaml

    pretrained model: google drive

  • NGM-v2

    experiment config: experiments/vgg16_ngmv2_willow.yaml

    pretrained model: google drive

  • NHGM-v2

    experiment config: experiments/vgg16_nhgmv2_willow.yaml

    pretrained model: google drive

  • NMGM-v2

    experiment config: experiments/vgg16_nmgmv2_willow.yaml

    pretrained model: google drive

model year remark Car Duck Face Motorbike Winebottle mean
NGM 2019 - 0.8420 0.7760 0.9940 0.7680 0.8830 0.8530
NHGM 2019 - 0.8650 0.7220 0.9990 0.7930 0.8940 0.8550
NMGM 2019 - 0.7850 0.9210 1.0000 0.7870 0.9480 0.8880
NGM-v2 2021 - 0.9740 0.9340 1.0000 0.9860 0.9830 0.9754
NHGM-v2 2021 - 0.9740 0.9390 1.0000 0.9860 0.9890 0.9780
NMGM-v2 2021 - 0.9760 0.9447 1.0000 1.0000 0.9902 0.9822

SPair-71k - 2GM

  • NGM

    experiment config: experiments/vgg16_ngm_spair71k.yaml

    pretrained model: google drive

  • NGM-v2

    experiment config: experiments/vgg16_ngmv2_spair71k.yaml

    pretrained model: google drive

  • NHGM-v2

    experiment config: experiments/vgg16_nhgmv2_spair71k.yaml

    pretrained model: google drive

model year aero bike bird boat bottle bus car cat chair cow dog horse mtbike person plant sheep train tv mean
NGM 2019 0.6644 0.5262 0.7696 0.4960 0.6766 0.7878 0.6764 0.6827 0.5917 0.7364 0.6391 0.6066 0.7074 0.6089 0.8754 0.6387 0.7979 0.9150 0.6887
NGM-v2 2021 0.6877 0.6331 0.8677 0.7013 0.6971 0.9467 0.8740 0.7737 0.7205 0.8067 0.7426 0.7253 0.7946 0.7340 0.9888 0.8123 0.9426 0.9867 0.8020
NHGM-v2 2021 0.6202 0.5781 0.8642 0.6846 0.6872 0.9335 0.8081 0.7656 0.6919 0.7987 0.6623 0.7171 0.7812 0.6953 0.9824 0.8444 0.9316 0.9926 0.7799

Run QAPLIB Experiments

Our proposed NGM is capable of handling pure combinatorial optimization problems from QAPLIB. We train one model on the problems with the same prefix (which means that they represent the same category of problems), and we set the objective function as the loss, resulting in a self-supervised learning scheme. To further elevate the performance on QAPLIB, we adopt randomly sampling based on Gumbel Sinkhorn.

About the Performance

The quality of solutions of our method is comparative and even better compared with the classic learning-free optimization method Sinkhorn-JA, and the inference time is significantly reduced. For more details, please visit the project page.

Only the NGM Model is Considered for QAPLIB

Since NGM-v2 only improves in image feature extractor which is not relevant to QAPLIB problems, we only test NGM on QAPLIB. Besides, the QAPLIB instances are equivalent to the formulation of two-graph matching problems, thus the hyper-graph and multi-graph variants are not considered for QAPLIB.

How to Run

For experiments on QAPLIB, run

python train_eval_qap.py --config experiments/ngm_qaplib.yaml

and the QAPLIB dataset shall be automatically downloaded. By default, training is run on the first category (bur) and testing is run on all datasets. You may modify the TRAIN.CLASS parameter for different problems.

NOTICE: you may need a large GPU memory (up to 48GB in our experiment) to run NGM on QAPLIB instances up to 150. Adjust the two parameters MAX_TRAIN_SIZE, MAX_TEST_SIZE accordingly to fit the size of your GPU memory.

File Organization

├── geo_edge_feature.py
|   the implementation of used geometric features 
├── gnn.py
|   the implementation of used graph neural networks 
├── hypermodel.py
|   the implementation of training/evaluation procedures of NHGM
├── hypermodel_v2.py
|   the implementation of training/evaluation procedures of NHGMv2
├── mgmmodel.py
|   the implementation of training/evaluation procedures of NMGM
├── model.py
|   the implementation of training/evaluation procedures of NGM (can handle both graph matching and QAP)
├── model_config.py
|   the declaration of model hyperparameters
├── model_v2.py
|   the implementation of training/evaluation procedures of NGMv2 (can only handle graph matching)
├── model_v2_linsat.py
|   the implementation of training/evaluation procedures of NGMv2 combined with LinSATNet
└── model_v2_topk.py
    the implementation of training/evaluation procedures of NGMv2 combined with topk-GM algorithm and partial matching handling AFA modules

some layers are borrowed from models/BBGM

Credits and Citation

Please cite the following paper if you use this model in your research:

@article{WangPAMI21,
  author = {Wang, Runzhong and Yan, Junchi and Yang, Xiaokang},
  title = {Neural Graph Matching Network: Learning Lawler's Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching},
  journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},
  year={2021}
}