Skip to main content

Convolutional Neural Networks

  • 1838 Accesses

Part of the Mathematics in Industry book series (MATHINDUSTRY,volume 37)

Abstract

A convolutional neural network (CNN, or ConvNet) is a class of deep neural networks, widely used for analyzing and processing images. Multilayer perceptrons, which we discussed in the previous chapter, usually require fully connected networks, where each neuron in one layer is connected to all neurons in the next layer. Unfortunately, this type of connections inescapably increases the number of weights.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-981-16-6046-7_7
  • Chapter length: 22 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   69.99
Price excludes VAT (USA)
  • ISBN: 978-981-16-6046-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   89.99
Price excludes VAT (USA)
Fig. 7.1
Fig. 7.2
Fig. 7.3
Fig. 7.4
Fig. 7.5
Fig. 7.6
Fig. 7.7
Fig. 7.8
Fig. 7.9
Fig. 7.10
Fig. 7.11
Fig. 7.12
Fig. 7.13
Fig. 7.14
Fig. 7.15
Fig. 7.16
Fig. 7.17
Fig. 7.18

References

  1. O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein et al., “ImageNet large scale visual recognition challenge,” International Journal of Computer Vision, vol. 115, no. 3, pp. 211–252, 2015.

    MathSciNet  CrossRef  Google Scholar 

  2. J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “ImageNet: A large-scale hierarchical image database,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition. IEEE, 2009, pp. 248–255.

    Google Scholar 

  3. A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems, 2012, pp. 1097–1105.

    Google Scholar 

  4. V. Vapnik, The nature of statistical learning theory. Springer Science & Business Media, 2013.

    Google Scholar 

  5. B. Schölkopf, A. J. Smola, F. Bach et al., Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, 2002.

    Google Scholar 

  6. D. H. Hubel and T. N. Wiesel, “Receptive fields of single neurones in the cat’s striate cortex,” The Journal of Physiology, vol. 148, no. 3, pp. 574–591, 1959.

    CrossRef  Google Scholar 

  7. Y. LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, W. Hubbard, and L. D. Jackel, “Backpropagation applied to handwritten zip code recognition,” Neural Computation, vol. 1, no. 4, pp. 541–551, 1989.

    CrossRef  Google Scholar 

  8. C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 1–9.

    Google Scholar 

  9. K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” arXiv preprint arXiv:1409.1556, 2014.

    Google Scholar 

  10. J. Johnson, A. Alahi, and L. Fei-Fei, “Perceptual losses for real-time style transfer and super-resolution,” in European Conference on Computer Vision (ECCV), 2016, pp. 694–711.

    Google Scholar 

  11. K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 770–778.

    Google Scholar 

  12. H. Li, Z. Xu, G. Taylor, C. Studer, and T. Goldstein, “Visualizing the loss landscape of neural nets,” in Advances in Neural Information Processing Systems, 2018, pp. 6389–6399.

    Google Scholar 

  13. J. C. Ye and W. K. Sung, “Understanding geometry of encoder-decoder CNNs,” in International Conference on Machine Learning, 2019, pp. 7064–7073.

    Google Scholar 

  14. Q. Nguyen and M. Hein, “Optimization landscape and expressivity of deep CNNs,” arXiv preprint arXiv:1710.10928, 2017.

    Google Scholar 

  15. G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely connected convolutional networks,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 4700–4708.

    Google Scholar 

  16. O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention. Springer, 2015, pp. 234–241.

    Google Scholar 

  17. K. H. Jin, M. T. McCann, E. Froustey, and M. Unser, “Deep convolutional neural network for inverse problems in imaging,” IEEE Transactions on Image Processing, vol. 26, no. 9, pp. 4509–4522, 2017.

    MathSciNet  CrossRef  Google Scholar 

  18. Y. Han and J. C. Ye, “Framing U-Net via deep convolutional framelets: Application to sparse-view CT,” IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 1418–1429, 2018.

    CrossRef  Google Scholar 

  19. S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” arXiv preprint arXiv:1502.03167, 2015.

    Google Scholar 

  20. J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM Journal on Imaging Sciences, vol. 11, no. 2, pp. 991–1048, 2018.

    MathSciNet  CrossRef  Google Scholar 

  21. J. Bruna and S. Mallat, “Invariant scattering convolution networks,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 8, pp. 1872–1886, 2013.

    CrossRef  Google Scholar 

  22. I. Goodfellow, Y. Bengio, and A. Courville, Deep learning. MIT Press, 2016.

    MATH  Google Scholar 

  23. N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” The Journal of Machine Learning Research, vol. 15, no. 1, pp. 1929–1958, 2014.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Ye, J.C. (2022). Convolutional Neural Networks. In: Geometry of Deep Learning. Mathematics in Industry, vol 37. Springer, Singapore. https://doi.org/10.1007/978-981-16-6046-7_7

Download citation