Skip to main content
SpringerLink
Log in

Geometric machine learning: research and applications

  • Published:
Multimedia Tools and Applications Aims and scope Submit manuscript

Abstract

Over the last decade, deep learning has revolutionized many traditional machine learning tasks, ranging from computer vision to natural language processing. Although deep learning has achieved excellent performance, it does not perform as well as expected on geometric (non-Euclidean domain) data. Recently, many studies on extending deep learning approaches for graphs and manifolds have merged. In this article, we aim to provide a comprehensive overview of geometric deep learning and comparative methods. First, we introduce the related work and history of the geometric deep learning field and the theoretical background. Next, we summarize the evaluation of the methods of graph and manifold. We further discuss the applications and benchmark datasets of these methods across various research domains. Finally, we propose potential research directions and challenges in this rapidly growing field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Abdi H, Williams LJ (2010) Principal component analysis. Wiley Interdiscip Rev Comput Stat 2(4):433–459

    Article  Google Scholar 

  2. Ahmed A, Shervashidze N, Narayanamurthy S, Josifovski V, Smola AJ (2013) Distributed large-scale natural graph factorization. In: Proceedings of the 22nd international conference on World Wide Web, pp 37–48

  3. Albishre K, Albathan M, Li Y (2015) Effective 20 newsgroups dataset cleaning. In: 2015 IEEE/WIC/ACM International Conference on Web Intelligence and Intelligent Agent Technology (WI-IAT), vol 3. IEEE, pp 98–101

  4. Atwood J, Towsley D (2016) Diffusion-convolutional neural networks. In: Advances in neural information processing systems, pp 1993–2001

  5. Bastings J, Titov I, Aziz W, Marcheggiani D, Sima’an K (2017) Graph convolutional encoders for syntax-aware neural machine translation. arXiv:1704.04675

  6. Battaglia PW, Hamrick JB, Bapst V, Sanchez-Gonzalez A, Zambaldi V, Malinowski M, Tacchetti A, Raposo D, Santoro A, Faulkner R et al (2018) Relational inductive biases, deep learning, and graph networks. arXiv:1806.01261

  7. Belkin M, Niyogi P (2002) Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in neural information processing systems, pp 585–591

  8. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 15(6):1373–1396

    Article  Google Scholar 

  9. Berg R, Kipf TN, Welling M (2017) Graph convolutional matrix completion. arXiv:1706.02263

  10. Blei DM, Ng AY, Jordan MI (2003) Latent dirichlet allocation. J Mach Learn Res 3:993–1022

    MATH  Google Scholar 

  11. Bo D, Wang X, Shi C, Shen H (2021) Beyond low-frequency information in graph convolutional networks. arXiv:2101.00797

  12. Bogo F, Romero J, Loper M, Black MJ (2014) Faust: Dataset and evaluation for 3d mesh registration. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 3794–3801

  13. Boscaini D, Masci J, Melzi S, Bronstein MM, Castellani U, Vandergheynst P (2015) Learning class-specific descriptors for deformable shapes using localized spectral convolutional networks. In: Computer Graphics Forum, vol 34. Wiley Online Library, pp 13–23

  14. Boscaini D, Masci J, Rodolà E, Bronstein M (2016) Learning shape correspondence with anisotropic convolutional neural networks. In: Advances in neural information processing systems, pp 3189–3197

  15. Bronstein MM, Bruna J, LeCun Y, Szlam A, Vandergheynst P (2017) Geometric deep learning: going beyond euclidean data. IEEE Signal Proc Mag 34(4):18–42

    Article  Google Scholar 

  16. Bruna J, Zaremba W, Szlam A, LeCun Y (2013) Spectral networks and locally connected networks on graphs. arXiv:1312.6203

  17. Buades A, Coll B, Morel J-M (2005) A non-local algorithm for image denoising. In: 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), vol 2. IEEE, pp 60–65

  18. Cao S, Lu W, Xu Q (2015) Grarep: Learning graph representations with global structural information. In: Proceedings of the 24th ACM international on conference on information and knowledge management, pp 891–900

  19. Cao W, Yan Z, He Z, He Z (2020) A comprehensive survey on geometric deep learning. IEEE Access 8:35929–35949

    Article  Google Scholar 

  20. Caragea C, Wu J, Ciobanu A, Williams K, Fernández-Ramírez J, Chen H-H, Wu Z, Giles L (2014) Citeseer x: A scholarly big dataset. In: European Conference on Information Retrieval. Springer, pp 311–322

  21. Chen J, Zhu J, Song L (2017) Stochastic training of graph convolutional networks with variance reduction. arXiv:1710.10568

  22. Chen J, Ma T, Xiao C (2018) Fastgcn: fast learning with graph convolutional networks via importance sampling. arXiv:1801.10247

  23. Chen M, Wei Z, Huang Z, Ding B, Li Y (2020) Simple and deep graph convolutional networks. arXiv:2007.02133

  24. Chen X, Li L-J, Fei-Fei L, Gupta A (2018) Iterative visual reasoning beyond convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 7239–7248

  25. Coley CW, Barzilay R, Green WH, Jaakkola TS, Jensen KF (2017) Convolutional embedding of attributed molecular graphs for physical property prediction. J Chem Inf Model 57(8):1757–1772

    Article  Google Scholar 

  26. Cui Z, Henrickson K, Ke R, Wang Y (2019) Traffic graph convolutional recurrent neural network: A deep learning framework for network-scale traffic learning and forecasting. IEEE Trans Intell Transp Syst

  27. De Cao N, Kipf T (2018) Molgan: An implicit generative model for small molecular graphs. arXiv:1805.119731805.11973

  28. Defferrard M, Bresson X, Vandergheynst P (2016) Convolutional neural networks on graphs with fast localized spectral filtering. In: Advances in neural information processing systems, pp 3844–3852

  29. Deng J, Dong W, Socher R, Li L-J, Li K, Fei-Fei L (2009) Imagenet: A large-scale hierarchical image database. In: 2009 IEEE conference on computer vision and pattern recognition. Ieee, pp 248–255

  30. Donoho DL, Grimes C (2003) Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci 100(10):5591–5596

    Article  MathSciNet  Google Scholar 

  31. Duvenaud DK, Maclaurin D, Iparraguirre J, Bombarell R, Hirzel T, Aspuru-Guzik A, Adams RP (2015) Convolutional networks on graphs for learning molecular fingerprints. In: Advances in neural information processing systems, pp 2224–2232

  32. Fan W, Ma Y, Li Q, He Y, Zhao E, Tang J, Yin D (2019) Graph neural networks for social recommendation. In: The World Wide Web Conference, pp 417–426

  33. Fout A, Byrd J, Shariat B, Ben-Hur A (2017) Protein interface prediction using graph convolutional networks. In: Advances in neural information processing systems, pp 6530–6539

  34. Gao H, Ji S (2019) Graph u-nets. arXiv:1905.05178

  35. Gao H, Wang Z, Ji S (2018) Large-scale learnable graph convolutional networks. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp 1416–1424

  36. Gilmer J, Schoenholz SS, Riley PF, Vinyals O, Dahl GE (2017) Neural message passing for quantum chemistry. arXiv:1704.01212

  37. Girshick R, Donahue J, Darrell T, Malik J (2014) Rich feature hierarchies for accurate object detection and semantic segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 580–587

  38. Goodfellow I, Bengio Y, Courville A, Bengio Y (2016) Deep learning, vol 1. MIT press Cambridge

  39. Gori M, Monfardini G, Scarselli F (2005) A new model for learning in graph domains. In: Proceedings. 2005 IEEE International Joint Conference on Neural Networks, 2005., vol 2. IEEE, pp 729–734

  40. Grover A, Leskovec J (2016) node2vec: Scalable feature learning for networks. In: Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp 855–864

  41. Guo M, Chou E, Huang D-A, Song S, Yeung S, Fei-Fei L (2018) Neural graph matching networks for fewshot 3d action recognition. In: Proceedings of the European conference on computer vision (ECCV), pp 653–669

  42. Hamaguchi T, Oiwa H, Shimbo M, Matsumoto Y (2017) Knowledge transfer for out-of-knowledge-base entities: A graph neural network approach. arXiv:1706.05674

  43. Hamilton W, Ying Z, Leskovec J (2017) Inductive representation learning on large graphs. In: Advances in neural information processing systems, pp 1024–1034

  44. Hammond DK, Vandergheynst P, Gribonval R (2011) Wavelets on graphs via spectral graph theory. Appl Comput Harmon Anal 30(2):129–150

    Article  MathSciNet  Google Scholar 

  45. Harper FM, Konstan JA (2015) The movielens datasets: History and context. Acm Trans Interactive Intell Syst (tiis) 5(4):1–19

    Google Scholar 

  46. He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770–778

  47. He K, Zhang X, Ren S, Sun J (2016) Identity mappings in deep residual networks. In: European conference on computer vision. Springer, pp 630–645

  48. Henaff M, Bruna J, LeCun Y (2015) Deep convolutional networks on graph-structured data. arXiv:1506.051631506.05163

  49. Hinton G, Deng L, Yu D, Dahl GE, Mohamed A-, Jaitly N, Senior A, Vanhoucke V, Nguyen P, Sainath TN et al (2012) Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups. IEEE Signal Process Mag 29(6):82–97

    Article  Google Scholar 

  50. Hjelm RD, Fedorov A, Lavoie-Marchildon S, Grewal K, Bachman P, Trischler A, Bengio Y (2018) Learning deep representations by mutual information estimation and maximization. arXiv:1808.06670

  51. Hu W, Liu B, Gomes J, Zitnik M, Liang P, Pande V, Leskovec J (2019) Strategies for pre-training graph neural networks. arXiv:1905.12265

  52. Huang G, Liu Z, Van Der Maaten L, Weinberger KQ (2017) Densely connected convolutional networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 4700–4708

  53. Huang Z, Wan C, Probst T, Van Gool L (2017) Deep learning on lie groups for skeleton-based action recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 6099–6108

  54. Jain A, Zamir AR, Savarese S, Saxena A (2016) Structural-rnn: Deep learning on spatio-temporal graphs. In: Proceedings of the ieee conference on computer vision and pattern recognition, pp 5308–5317

  55. Jin X-B, Yu X-H, Su T-L, Yang D-N, Bai Y-T, Kong J-L, Wang L (2021) Distributed deep fusion predictor for a multi-sensor system based on causality entropy. Entropy 23(2)

  56. Kim D, Oh A (2021) How to find your friendly neighborhood: Graph attention design with self-supervision. In: International Conference on Learning Representations

  57. Kipf TN, Welling M (2016) Semi-supervised classification with graph convolutional networks. arXiv:1609.02907

  58. Kipf TN, Welling M (2016) Variational graph auto-encoders. arXiv:1611.07308

  59. Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet classification with deep convolutional neural networks. In: Advances in neural information processing systems, pp 1097–1105

  60. Ktena SI, Parisot S, Ferrante E, Rajchl M, Lee M, Glocker B, Rueckert D (2017) Distance metric learning using graph convolutional networks: Application to functional brain networks. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, pp 469–477

  61. LeCun Y (1998) The mnist database of handwritten digits. http://yann.lecun.com/exdb/mnist/

  62. LeCun Y, Bengio Y et al (1995) Convolutional networks for images, speech, and time series. Handb Brain Theory Neural Netw 3361(10):1995

    Google Scholar 

  63. LeCun Y, Bengio Y, Hinton G (2015) Deep learning. Nature 521(7553):436–444

    Article  Google Scholar 

  64. LeCun Y, Bottou L, Bengio Y, Haffner P (1998) Gradient-based learning applied to document recognition. Proc IEEE 86(11):2278–2324

    Article  Google Scholar 

  65. Lee JB, Rossi RA, Kim S, Ahmed NK, Koh E (2019) Attention models in graphs: A survey. ACM Trans Knowl Discov Data (TKDD) 13(6):1–25

    Article  Google Scholar 

  66. Lee J, Lee I, Kang J (2019) Self-attention graph pooling. arXiv:1904.08082

  67. Levie R, Monti F, Bresson X, Bronstein MM (2018) Cayleynets: Graph convolutional neural networks with complex rational spectral filters. IEEE Trans Signal Process 67(1):97–109

    Article  MathSciNet  Google Scholar 

  68. Li G, Muller M, Thabet A, Ghanem B (2019) Deepgcns: Can gcns go as deep as cnns?. In: Proceedings of the IEEE International Conference on Computer Vision, pp 9267–9276

  69. Li R, Wang S, Zhu F, Huang J (2018) Adaptive graph convolutional neural networks. arXiv:1801.03226

  70. Li Y, Cao W (2019) An extended multilayer perceptron model using reduced geometric algebra. IEEE Access 7:129815–129823

    Article  Google Scholar 

  71. Litany O, Remez T, Rodola E, Bronstein A, Bronstein M (2017) Deep functional maps: Structured prediction for dense shape correspondence. In: Proceedings of the IEEE International Conference on Computer Vision, pp 5659–5667

  72. Liu Z, Zhou J (2020) Introduction to graph neural networks. Synth Lect Artif Intell Mach Learn 14(2):1–127

    MathSciNet  MATH  Google Scholar 

  73. Liu Z, Chen C, Li L, Zhou J, Li X, Song L, Qi Y (2019) Geniepath: Graph neural networks with adaptive receptive paths. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol 33, pp 4424–4431

  74. Looks M, Herreshoff M, Hutchins D, Norvig P (2017) Deep learning with dynamic computation graphs. arXiv:1702.02181

  75. Lovász L et al (1993) Random walks on graphs: A survey. Comb Paul Erdos Eighty 2(1):1–46

    Google Scholar 

  76. Ma Y, Guo Z, Ren Z, Tang J, Yin D (2020) Streaming graph neural networks. In: Proceedings of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval, pp 719–728

  77. Masci J, Boscaini D, Bronstein M, Vandergheynst P (2015) Geodesic convolutional neural networks on riemannian manifolds. In: Proceedings of the IEEE international conference on computer vision workshops, pp 37–45

  78. McCallum A (2017) Cora dataset

  79. The Princeton ModelNet. https://modelnet.cs.princeton.edu/. Online; accessed: October 2019

  80. Monti F, Boscaini D, Masci J, Rodola E, Svoboda J, Bronstein MM (2017) Geometric deep learning on graphs and manifolds using mixture model cnns. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 5115–5124

  81. Monti F, Bronstein M, Bresson X (2017) Geometric matrix completion with recurrent multi-graph neural networks. In: Advances in Neural Information Processing Systems, pp 3697–3707

  82. Narasimhan M, Lazebnik S, Schwing A (2018) Out of the box: Reasoning with graph convolution nets for factual visual question answering. In: Advances in neural information processing systems, pp 2654–2665

  83. Niepert M, Ahmed M, Kutzkov K (2016) Learning convolutional neural networks for graphs. In: International conference on machine learning, pp 2014–2023

  84. Ou M, Cui P, Pei J, Zhang Z, Zhu W (2016) Asymmetric transitivity preserving graph embedding. In: Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp 1105–1114

  85. Pan S, Hu R, Long G, Jiang J, Yao L, Zhang C (2018) Adversarially regularized graph autoencoder for graph embedding. arXiv:1802.04407

  86. Parisot S, Ktena SI, Ferrante E, Lee M, Moreno RG, Glocker B, Rueckert D (2017) Spectral graph convolutions for population-based disease prediction. In: International conference on medical image computing and computer-assisted intervention. Springer, pp 177–185

  87. Perozzi B, Al-Rfou R, Skiena S (2014) Deepwalk: Online learning of social representations. In: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp 701–710

  88. Pham T, Tran T, Phung D, Venkatesh S (2016) Column networks for collective classification. arXiv:1609.04508

  89. Qi CR, Su H, Mo K, Guibas LJ (2017) Pointnet: Deep learning on point sets for 3d classification and segmentation. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 652–660

  90. Qi CR, Yi L, Su H, Guibas LJ (2017) Pointnet++: Deep hierarchical feature learning on point sets in a metric space. In: Advances in neural information processing systems, pp 5099–5108

  91. Qi X, Liao R, Jia J, Fidler S, Urtasun R (2017) 3d graph neural networks for rgbd semantic segmentation. In: Proceedings of the IEEE International Conference on Computer Vision, pp 5199–5208

  92. Rahimi A, Cohn T, Baldwin T (2018) Semi-supervised user geolocation via graph convolutional networks. arXiv:1804.08049

  93. Ren S, He K, Girshick R, Sun J (2015) Faster r-cnn: Towards real-time object detection with region proposal networks. In: Advances in neural information processing systems, pp 91–99

  94. Rhee S, Seo S, Kim S (2017) Hybrid approach of relation network and localized graph convolutional filtering for breast cancer subtype classification. arXiv:1711.05859

  95. Rong Y, Huang W, Xu T, Huang J (2019) Dropedge: Towards deep graph convolutional networks on node classification. In: International Conference on Learning Representations

  96. Ronneberger O, Fischer P, Brox T (2015) U-net: Convolutional networks for biomedical image segmentation. In: International Conference on Medical image computing and computer-assisted intervention. Springer, pp 234–241

  97. Roweis ST, Saul LK (2000) Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500):2323–2326

    Article  Google Scholar 

  98. Ruder S (2016) An overview of gradient descent optimization algorithms. arXiv:1609.04747

  99. Santoro A, Raposo D, Barrett DG, Malinowski M, Pascanu R, Battaglia P, Lillicrap T (2017) A simple neural network module for relational reasoning. In: Advances in neural information processing systems, pp 4967–4976

  100. Scarselli F, Gori M, Tsoi AC, Hagenbuchner M, Monfardini G (2008) The graph neural network model. IEEE Trans Neural Netw 20(1):61–80

    Article  Google Scholar 

  101. Schlichtkrull M, Kipf TN, Bloem P, Van Den Berg R, Titov I, Welling M (2018) Modeling relational data with graph convolutional networks. In: European Semantic Web Conference. Springer, pp 593–607

  102. Shaw B, Jebara T (2009) Structure preserving embedding. In: Proceedings of the 26th Annual International Conference on Machine Learning, pp 937–944

  103. Shchur O, Günnemann S (2019) Overlapping community detection with graph neural networks. arXiv:1909.12201

  104. Sherstinsky A (2020) Fundamentals of recurrent neural network (rnn) and long short-term memory (lstm) network. Physica D: Nonlinear Phenom 404:132306

    Article  MathSciNet  Google Scholar 

  105. Shi C, Li Y, Zhang J, Sun Y, Philip SY (2016) A survey of heterogeneous information network analysis. IEEE Trans Knowl Data Eng 29(1):17–37

    Article  Google Scholar 

  106. Shuman DI, Narang SK, Frossard P, Ortega A, Vandergheynst P (2013) The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Process Mag 30(3):83–98

    Article  Google Scholar 

  107. Su H, Maji S, Kalogerakis E, Learned-Miller E (2015) Multi-view convolutional neural networks for 3d shape recognition. In: Proceedings of the IEEE international conference on computer vision, pp 945–953

  108. Sutskever I, Vinyals O, Le QV (2014) Sequence to sequence learning with neural networks. In: Advances in neural information processing systems, pp 3104–3112

  109. Szegedy C, Vanhoucke V, Ioffe S, Shlens J, Wojna Z (2016) Rethinking the inception architecture for computer vision. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 2818–2826

  110. Tailor SA, Opolka FL, Liò P, Lane ND (2021) Adaptive filters and aggregator fusion for efficient graph convolutions. arXiv:2104.01481

  111. Tang J, Qu M, Mei Q (2015) Pte: Predictive text embedding through large-scale heterogeneous text networks. In: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp 1165–1174

  112. Tang J, Qu M, Wang M, Zhang M, Yan J, Mei Q (2015) Line: Large-scale information network embedding. In: Proceedings of the 24th international conference on world wide web, pp 1067–1077

  113. Tang J, Gao H, Liu H, Das Sarma A (2012) etrust: Understanding trust evolution in an online world. In: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, pp 253–261

  114. Tenenbaum JB, De Silva V, Langford JC (2000) A global geometric framework for nonlinear dimensionality reduction. Science 290(5500):2319–2323

    Article  Google Scholar 

  115. Thekumparampil KK, Wang C, Oh S, Li L-J (2018) Attention-based graph neural network for semi-supervised learning. arXiv:1803.03735

  116. Tu K, Cui P, Wang X, Yu PS, Zhu W (2018) Deep recursive network embedding with regular equivalence. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp 2357–2366

  117. Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser L, Polosukhin I (2017) Attention is all you need. In: Advances in neural information processing systems, pp 5998–6008

  118. Veličković P, Cucurull G, Casanova A, Romero A, Lio P, Bengio Y (2017) Graph attention networks. arXiv:1710.10903

  119. Wang D, Cui P, Zhu W (2016) Structural deep network embedding. In: Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discovery and data mining, pp 1225–1234

  120. Wang R, Shen M, Cao W (2019) Multivector sparse representation for multispectral images using geometric algebra. IEEE Access 7:12755–12767

    Article  Google Scholar 

  121. Wang X, Girshick R, Gupta A, He K (2018) Non-local neural networks. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 7794–7803

  122. Wang Z, Lv Q, Lan X, Zhang Y (2018) Cross-lingual knowledge graph alignment via graph convolutional networks. In: Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp 349–357

  123. Wilensky U, Reisman K (2006) Thinking like a wolf, a sheep, or a firefly: Learning biology through constructing and testing computational theories-an embodied modeling approach. Cogn Instruct 24(2):171–209

    Article  Google Scholar 

  124. Wu Z, Song S, Khosla A, Yu F, Zhang L, Tang X, Xiao J (2015) 3d shapenets: A deep representation for volumetric shapes. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 1912–1920

  125. Xu K, Li C, Tian Y, Sonobe T, Kawarabayashi K-I, Jegelka S (2018) Representation learning on graphs with jumping knowledge networks. arXiv:1806.03536

  126. Yan S, Xiong Y, Lin D (2018) Spatial temporal graph convolutional networks for skeleton-based action recognition. arXiv:1801.07455

  127. Yi L, Su H, Guo X, Guibas LJ (2017) Syncspeccnn: Synchronized spectral cnn for 3d shape segmentation. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 2282–2290

  128. Ying Z, You J, Morris C, Ren X, Hamilton W, Leskovec J (2018) Hierarchical graph representation learning with differentiable pooling. In: Advances in neural information processing systems, pp 4800–4810

  129. Yu F, Koltun V (2015) Multi-scale context aggregation by dilated convolutions. arXiv:1511.07122

  130. Wu Z, Pan S, Chen F, Long G, Zhang C, Philip S Y (2020) A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems

  131. Zhang J, Shi X, Xie J, Ma H, King I, Yeung D-Y (2018) Gaan: Gated attention networks for learning on large and spatiotemporal graphs. arXiv:1803.07294

  132. Zhang J, Shi X, Zhao S, King I (2019) Star-gcn: Stacked and reconstructed graph convolutional networks for recommender systems. arXiv:1905.13129

  133. Zhang S, Yin H, Chen T, Hung QVN, Huang Z, Cui L (2020) Gcn-based user representation learning for unifying robust recommendation and fraudster detection. arXiv:2005.10150

  134. Zhang Z, Li M, Lin X, Wang Y, He F (2019) Multistep speed prediction on traffic networks: A deep learning approach considering spatio-temporal dependencies. Transp Res Part C: Emerging Technol 105:297–322

    Article  Google Scholar 

  135. Zhang Z, Zha H (2003) Nonlinear dimension reduction via local tangent space alignment. In: International Conference on Intelligent Data Engineering and Automated Learning. Springer, pp 477–481

  136. Zhang Z, Cui P, Zhu W (2020) Deep learning on graphs: A survey. IEEE Trans Knowl Data Eng

  137. Zhou J, Cui G, Zhang Z, Yang C, Liu Z, Wang L, Li C, Sun M (2018) Graph neural networks: A review of methods and applications. arXiv:1812.08434

  138. Zilly JG, Srivastava RK, Koutnık J, Schmidhuber J (2017) Recurrent highway networks. In: International Conference on Machine Learning, pp 4189–4198

  139. Zitnik M, Agrawal M, Leskovec J (2018) Modeling polypharmacy side effects with graph convolutional networks. Bioinformatics 34(13):i457–i466

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 61771322 and Grant 61871186 and in part by the Fundamental Research Foundation of Shenzhen under Grant JCYJ20190808160815125.

Author information

Authors and Affiliations

  1. College of Electronics and Information Engineering, Shenzhen University, Shenzhen, 518060, China

    Wenming Cao, Canta Zheng, Zhiyue Yan & Weixin Xie

  2. Video Processing and Communication Laboratory, Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO, 65211, USA

    Zhihai He

Authors
  1. Wenming Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Canta Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhiyue Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhihai He
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Weixin Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Canta Zheng or Weixin Xie.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, W., Zheng, C., Yan, Z. et al. Geometric machine learning: research and applications. Multimed Tools Appl 81, 30545–30597 (2022). https://doi.org/10.1007/s11042-022-12683-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11042-022-12683-9

Keywords

Search

Navigation