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Revisiting convolutional neural network on graphs with polynomial approximations of Laplace–Beltrami spectral filtering

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Abstract

This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace–Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We define spectral filters via the LB operator on a graph and explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters. We then update the LB operator for pooling in the LB-CNN. We employ the brain image data from Alzheimer’s Disease Neuroimaging Initiative (ADNI) and Open Access Series of Imaging Studies (OASIS) to demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of two datasets, we showed that the LB-CNN slightly improves classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. The LB-CNN trained via the ADNI dataset can achieve reasonable classification accuracy for the OASIS dataset. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).

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Availability of data and material

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.ucla.edu).

Code availability

https://bieqa.github.io/deeplearning.html.

References

  1. Apostolova LG, Dinov ID, Dutton RA, Hayashi KM, Toga AW, Cummings JL, Thompson PM (2006) 3D comparison of hippocampal atrophy in amnestic mild cognitive impairment and alzheimer’s disease. Brain 129:2867–2873

    Article  Google Scholar 

  2. Atwood J, Towsley D (2015) Diffusion-convolutional neural networks. arXiv preprint arXiv:151102136

  3. Basaia S, Agosta F, Wagner L, Canu E, Magnani G, Santangelo R, Filippi M, Initiative ADN, et al. (2019) Automated classification of alzheimer’s disease and mild cognitive impairment using a single MRI and deep neural networks. NeuroImage Clin 21:101645

  4. Bianchi FM, Grattarola D, Alippi C, Livi L (2019) Graph neural networks with convolutional arma filters. arXiv preprint arXiv:190101343

  5. 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. Comput Graph Forum 34(5):13–23

    Article  Google Scholar 

  6. Boscaini D, Masci J, Rodoia E, Bronstein M (2016a) Learning shape correspondence with anisotropic convolutional neural networks. In: NIPS’16 Proceedings of the 30th international conference on neural information processing systems, ACM, pp 3197–3205

  7. Boscaini D, Masci J, Rodola E, Bronstein MM, Cremers D (2016b) Anisotropic diffusion descriptors. Comput Graph Forum 35(2):431–441

    Article  Google Scholar 

  8. Bronstein M, Bruna J, LeCun Y, Szlam A, Vandergheynst P (2017) Geometric deep learning: going beyond Euclidean data. IEEE Signal Process Mag 34(4):18–42

    Article  Google Scholar 

  9. Bruna J, Zaremba W, Szlam A, LeCun Y (2013) Spectral networks and locally connected networks on graphs. arXiv preprint arXiv:13126203

  10. Chung M, Taylor J (2004) Diffusion smoothing on brain surface via finite element method. Proc IEEE Int Symp Biomed Imaging (ISBI) 1:432–435

    Google Scholar 

  11. Chung M, Qiu A, Seo S, Vorperian H (2015) Unified heat kernel regression for diffusion, kernel smoothing and wavelets on manifolds and its application to mandible growth modeling in CT images. Med Image Anal 22:63–76

    Article  Google Scholar 

  12. Coifman RR, Maggioni M (2006) Diffusion wavelets. Appl Comput Harmonic Anal 21(1):53–94

    Article  MathSciNet  Google Scholar 

  13. Cuingnet R, Gerardin E, Tessieras J, Auzias G, Lehéricy S, Habert MO, Chupin M, Benali H, Colliot O (2011) Automatic classification of patients with Alzheimer’s disease from structural MRI: a comparison of ten methods using the ADNI database. Neuroimage 56(2):766–781

    Article  Google Scholar 

  14. Defferrard M, Bresson X, Vandergheynst P (2016a) Convolutional neural networks on graphs with fast localized spectral filtering. In: Proceedings of the 30th international conference on neural information processing systems, NIPS, pp 3844–3852

  15. Defferrard M, Bresson X, Vandergheynst P (2016b) Convolutional neural networks on graphs with fast localized spectral filtering. In: NIPS’16 Proceedings of the 30th international conference on neural information processing systems, ACM, pp 3844–3852

  16. Dhillon IS, Guan Y, Kulis B (2007) Weighted graph cuts without eigenvectors a multilevel approach. IEEE Trans Pattern Anal Mach Intell 29(11):1944–1957

    Article  Google Scholar 

  17. Du J, Younes L, Qiu A (2011) Whole brain diffeomorphic metric mapping via integration of sulcal and gyral curves, cortical surfaces, and images. NeuroImage 56(1):162–173

    Article  Google Scholar 

  18. Duvenaud DK, Maclaurin D, Aguilera-Iparraguirre J, Gomez-Bombarelli R, Hirzel T, Aspuru-Guzik A, Adams RP (2015) Convolutional networks on graphs for learning molecular fingerprints. arXiv preprint arXiv:150909292

  19. Fan Y, Gur R, Gur R, Wu X, Shen D, Calkins M, Davatzikos C (2008) Unaffected family members and schizophrenia patients share brain structure patterns: a high-dimensional pattern classification study. Biol Psychiatry 63(1):118–124

    Article  Google Scholar 

  20. Fischl B, Salat DH, Busa E, Albert M, Dieterich M, Haselgrove C, Van Der Kouwe A, Killiany R, Kennedy D, Klaveness S et al (2002) Whole brain segmentation: automated labeling of neuroanatomical structures in the human brain. Neuron 33(3):341–355

    Article  Google Scholar 

  21. Gilmer J, Schoenholz SS, Riley PF, Vinyals O, Dahl GE (2017) Neural message passing for quantum chemistry. arXiv preprint arXiv:170401212

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

    Article  MathSciNet  Google Scholar 

  23. Henaff M, Bruna J, LeCun Y (2015) Deep convolutional networks on graph-structured data. arXiv preprint arXiv:150605163

  24. Hosseini-Asl E, Keynton R, El-Baz A (2016) Alzheimer’s disease diagnostics by adaptation of 3d convolutional network. In: 2016 IEEE international conference on image processing (ICIP). IEEE, pp 126–130

  25. Huang SG, Lyu I, Qiu A, Chung M (2019) Fast polynomial approximation of heat diffusion on manifolds and its application to brain sulcal and gyral graph pattern analysis. IEEE Transactions on Medical Imaging, pp under 2nd review. arXiv:1911.02721

  26. Huang SG, Lyu I, Qiu A, Chung MK (2020) Fast polynomial approximation of heat kernel convolution on manifolds and its application to brain sulcal and gyral graph pattern analysis. IEEE Trans Med Imaging 39(6):2201–2212

    Article  Google Scholar 

  27. Islam J, Zhang Y (2018) Brain mri analysis for alzheimer’s disease diagnosis using an ensemble system of deep convolutional neural networks. Brain Inform 5(2):2

    Article  Google Scholar 

  28. Isufi E, Loukas A, Simonetto A, Leus G (2017) Filtering random graph processes over random time-varying graphs. IEEE Trans Signal Process 65(16):4406–4421

    Article  MathSciNet  Google Scholar 

  29. James G, Witten D, Hastie T, Tibshirani R (2013) An introduction to statistical learning, vol 112. Springer, Berlin

    Book  Google Scholar 

  30. Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359–392

    Article  MathSciNet  Google Scholar 

  31. Kim WH, Pachauri D, Hatt C, Chung MK, Johnson S, Singh V (2012) Wavelet based multi-scale shape features on arbitrary surfaces for cortical thickness discrimination. In: Advances in neural information processing systems, pp 1241–1249

  32. Kipf TN, Welling M (2016) Semi-supervised classification with graph convolutional networks. arXiv preprint arXiv:160902907

  33. Klicpera J, Bojchevski A, Günnemann S (2018) Predict then propagate: graph neural networks meet personalized pagerank. arXiv preprint arXiv:181005997

  34. Korolev S, Safiullin A, Belyaev M, Dodonova Y (2017) Residual and plain convolutional neural networks for 3d brain MRI classification. In: 2017 IEEE 14th international symposium on biomedical imaging (ISBI 2017). IEEE, pp 835–838

  35. 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. arXiv preprint arXiv:170302161

  36. 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 

  37. Li Y, Tarlow D, Brockschmidt M, Zemel R (2015) Gated graph sequence neural networks. arXiv preprint arXiv:151105493

  38. Liu M, Zhang J, Adeli E, Shen D (2018) Landmark-based deep multi-instance learning for brain disease diagnosis. Med Image Anal 43:157–168

    Article  Google Scholar 

  39. Liu X, Tosun D, Weiner MW, Schuff N, Initiative ADN et al (2013) Locally linear embedding (lle) for MRI based alzheimer’s disease classification. Neuroimage 83:148–157

    Article  Google Scholar 

  40. Loukas A, Simonetto A, Leus G (2015) Distributed autoregressive moving average graph filters. IEEE Signal Process Lett 22(11):1931–1935

    Article  Google Scholar 

  41. Masci J, Boscaini D, Bronstein MM, Vandergheynst P (2015) Geodesic convolutional neural networks on riemannian manifolds. In: 2015 IEEE international conference on computer vision (ICCV). IEEE, pp 832–840

  42. Meyer M, Desbrun M, Schröder P, Barr AH (2003) Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and mathematics III, Springer, pp 35–57

  43. Monti F, Boscaini D, Masci J, Rodolà E, Svoboda J, Bronstein MM (2016) Geometric deep learning on graphs and manifolds using mixture model cnns. arXiv preprint arXiv:161108402

  44. Niepert M, Ahmed M, Kutzkov K (2016) Learning convolutional neural networks for graphs. In: Proceeding of the 33rd international conference on machine learning. ACM, p 2014–2023

  45. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (2010) NIST handbook of mathematical functions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  46. Perozzi B, Al-Rfou R, Skiena S (2014) Deepwalk: online learning of social representations. In: Proceedings of the 20th ACM SIGKDD, ACM, pp 701–710

  47. Perrault-Joncas DC, Meilǎ M, McQueen J (2017) Improved graph Laplacian via geometric consistency. In: Proceedings of the 31st international conference on neural information processing systems, pp 4460–4469

  48. Qiu A, Bitouk D, Miller M (2006) Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace–Beltrami operator. IEEE Trans Med Imaging 25:1296–1396

    Article  Google Scholar 

  49. Qiu A, Fennema-Notestine C, Dale A, Miller M, the Alzheimer’s Disease Neuroimaging Initiative (2009) Regional shape abnormalities in mild cognitive impairment and Alzheimer’s disease. Neuroimage 45:656–661

  50. Shi J, Malik J (2000) Normalized cuts and image segmentation. IEEE Trans Pattern Anal Mach Intell 22(8):888–905

    Article  Google Scholar 

  51. Shuman DI, Ricaud B, Vandergheynst P (2016) Vertex-frequency analysis on graphs. Appl Comput Harmonic Anal 40(2):260–291

    Article  MathSciNet  Google Scholar 

  52. Tan M, Qiu A (2015) Spectral Laplace–Beltrami wavelets with applications in medical images. IEEE Trans Med Imaging 34:1005–1017

    Article  Google Scholar 

  53. Wee CY, Liu C, Lee A, Poh JS, Ji H, Qiu A, Initiative ADN (2019) Cortical graph neural network for AD and MCI diagnosis and transfer learning across populations. NeuroImage Clin 23:101929

  54. Wijesinghe WAS, Wang Q (2019) Dfnets: Spectral cnns for graphs with feedback-looped filters. In: Advances in neural information processing systems, pp 6009–6020

  55. Yang X, Goh A, Chen S, Qiu A (2013) Evolution of hippocampal shapes across the human lifespan. Hum Brain Mapp 34:3075–3085

    Article  Google Scholar 

  56. Yi L, Su H, Guo X, Guibas L (2017) Syncspeccnn: Synchronized spectral cnn for 3d shape segmentation. In: Computer vision and pattern recognition (CVPR), conference on. IEEE, pp 6584–6592

  57. Zhang Z, Cui P, Zhu W (2020) Deep learning on graphs: a survey. IEEE Trans Knowl Data Eng. Preprint. https://doi.org/10.1109/TKDE.2020.2981333

  58. Zhong J, Phua DYL, Qiu A (2010) Quantitative evaluation of lddmm, freesurfer, and caret for cortical surface mapping. Neuroimage 52(1):131–141

    Article  Google Scholar 

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Funding

This research/project is supported by the National Science Foundation MDS-2010778, National Institute of Health R01 EB022856, EB02875, and National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG-GC-2019-002). Additional funding is provided by the Singapore Ministry of Education (Academic research fund Tier 1; NUHSRO/2017/052/T1-SRP-Partnership/01), NUS Institute of Data Science. This research was also supported by the A*STAR Computational Resource Centre through the use of its high-performance computing facilities.

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Authors and Affiliations

  1. Department of Biomedical Engineering, National University of Singapore, Singapore, 117583, Singapore

    Shih-Gu Huang

  2. Department of Biostatistics and Medical Informatics, University of Wisconsin, Madison, WI, 53706, USA

    Moo K. Chung

  3. Department of Biomedical Engineering, The N.1 Institute for Health and Institute of Data Science, National University of Singapore, Singapore, 117583, Singapore

    Anqi Qiu

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  1. Shih-Gu Huang
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Correspondence to Anqi Qiu.

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Huang, SG., Chung, M.K., Qiu, A. et al. Revisiting convolutional neural network on graphs with polynomial approximations of Laplace–Beltrami spectral filtering. Neural Comput & Applic 33, 13693–13704 (2021). https://doi.org/10.1007/s00521-021-06006-6

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