Abstract
Several techniques were designed during last few years to improve the performance of deep architecture by means of appropriate loss functions or activation functions. Arguably, softmax is the traditionally convenient to train Deep Convolutional Neural Networks (DCNNs) for classification task. However, the modern deep learning architectures have exposed its limitation towards feature discriminability. In this paper, we offered a supervision signal for discriminative image features through a modification in softmax to boost up the power of loss function. Amending the original softmax loss and motivated by the A-softmax loss for face recognition, we fixed the angular margin to introduce a unit margin softmax loss. The improved alternative form of softmax is trainable, easy to optimize and stable for usage along with Stochastic Gradient Descent (SGD) and Laplacian Smoothing Stochastic Gradient Descent (LS-SGD) and applicable to classify the digits in image. Experimental results demonstrate a state-of-the-art performance on famous database of handwritten digits the Modified National Institute of Standards and Technology (MNIST) database.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Agarwal, S., Terrail, J.O.D., Jurie, F.: Recent advances in object detection in the age of deep convolutional neural networks. arXiv preprint arXiv:1809.03193 (2018)
Ashiquzzaman, A., Tushar, A.K.: Handwritten Arabic numeral recognition using deep learning neural networks. In: 2017 IEEE International Conference on Imaging, Vision & Pattern Recognition (icIVPR), pp. 1–4. IEEE (2017)
Ba, J., Mnih, V., Kavukcuoglu, K.: Multiple object recognition with visual attention. arXiv preprint arXiv:1412.7755 (2014)
Bhatia, E.N.: Optical character recognition techniques: a review. Int. J. Adv. Res. Comput. Sci. Softw. Eng. 4(5) (2014)
Bottou, L.: Large-scale machine learning with stochastic gradient descent. In: Lechevallier, Y., Saporta, G. (eds.) Proceedings of COMPSTAT’2010, pp. 177–186. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-7908-2604-3_16
Bottou, L., Curtis, F.E., Nocedal, J.: Optimization methods for large-scale machine learning. SIAM Rev. 60(2), 223–311 (2018)
Chaudhari, P., Oberman, A., Osher, S., Soatto, S., Carlier, G.: Deep relaxation: partial differential equations for optimizing deep neural networks. Res. Math. Sci. 5(3), 1–30 (2018). https://doi.org/10.1007/s40687-018-0148-y
Defazio, A., Bach, F., Lacoste-Julien, S.: SAGA: a fast incremental gradient method with support for non-strongly convex composite objectives. In: Advances in Neural Information Processing Systems, pp. 1646–1654 (2014)
Deng, J., Guo, J., Xue, N., Zafeiriou, S.: ArcFace: additive angular margin loss for deep face recognition. arXiv preprint arXiv:1801.07698 (2018)
Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)
Goodfellow, I.J., Warde-Farley, D., Mirza, M., Courville, A., Bengio, Y.: Maxout networks. arXiv preprint arXiv:1302.4389 (2013)
Jarrett, K., Kavukcuoglu, K., LeCun, Y., et al.: What is the best multi-stage architecture for object recognition? In: 2009 IEEE 12th International Conference on Computer Vision, pp. 2146–2153. IEEE (2009)
Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction. In: Advances in Neural Information Processing Systems, pp. 315–323 (2013)
Laval, J.A., Leclercq, L.: The Hamilton-Jacobi partial differential equation and the three representations of traffic flow. Transp. Res. Part B Methodol. 52, 17–30 (2013)
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436 (2015)
LeCun, Y., Cortes, C., Burges, C.J.: The MNIST database of handwritten digits, 1998, vol. 10, p. 34 (1998). http://yann.lecun.com/exdb/mnist
Lee, C.Y., Gallagher, P.W., Tu, Z.: Generalizing pooling functions in convolutional neural networks: mixed, gated, and tree. In: Artificial Intelligence and Statistics, pp. 464–472 (2016)
Liu, C.L., Sako, H., Fujisawa, H.: Discriminative learning quadratic discriminant function for handwriting recognition. IEEE Trans. Neural Netw. 15(2), 430–444 (2004)
Liu, W., Wen, Y., Yu, Z., Li, M., Raj, B., Song, L.: SphereFace: deep hypersphere embedding for face recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 212–220 (2017)
Liu, W., Wen, Y., Yu, Z., Yang, M.: Large-margin softmax loss for convolutional neural networks. In: ICML, vol. 2. p. 7 (2016)
Osher, S., Wang, B., Yin, P., Luo, X., Pham, M., Lin, A.: Laplacian smoothing gradient descent. arXiv preprint arXiv:1806.06317 (2018)
Ranjan, R., Castillo, C.D., Chellappa, R.: L2-constrained softmax loss for discriminative face verification. arXiv preprint arXiv:1703.09507 (2017)
Ren, S., He, K., Girshick, R., Sun, J.: Faster R-CNN: towards real-time object detection with region proposal networks. In: Advances in Neural Information Processing Systems, pp. 91–99 (2015)
Romero, A., Ballas, N., Kahou, S.E., Chassang, A., Gatta, C., Bengio, Y.: FitNets: hints for thin deep nets. arXiv preprint arXiv:1412.6550 (2014)
Ruder, S.: An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747 (2016)
Schmidhuber, J.: Deep learning in neural networks: an overview. Neural Netw. 61, 85–117 (2015)
Schroff, F., Kalenichenko, D., Philbin, J.: FaceNet: a unified embedding for face recognition and clustering. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 815–823 (2015)
Suleman, M., Lu, D., Yue, C., Ul Rahman, J., Anjum, N.: He-Laplace method for general nonlinear periodic solitary solution of vibration equations. J. Low Freq. Noise Vib. Act. Control. 38, 1297–1304 (2019). https://doi.org/10.1177/1461348418816266
Sun, Y., Chen, Y., Wang, X., Tang, X.: Deep learning face representation by joint identification-verification. In: Advances in Neural Information Processing Systems, pp. 1988–1996 (2014)
Ul Rahman, J., Chen, Q., Yang, Z.: Additive parameter for deep face recognition. Commun. Math. Stat., 1–15 (2019)
Ul Rahman, J., Suleman, M., Lu, D., He, J.H., Ramzan, M.: He-Elzaki method for spatial diffusion of biological population. Fractals (2009)
Voulodimos, A., Doulamis, N., Doulamis, A., Protopapadakis, E.: Deep learning for computer vision: a brief review. Comput. Intell. Neurosci. (2018)
Wan, L., Zeiler, M., Zhang, S., Le Cun, Y., Fergus, R.: Regularization of neural networks using dropconnect. In: International Conference on Machine Learning, pp. 1058–1066 (2013)
Wang, H., et al.: CosFace: large margin cosine loss for deep face recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5265–5274 (2018)
Zhang, Q., Yang, L.T., Chen, Z., Li, P.: A survey on deep learning for big data. Inf. Fusion 42, 146–157 (2018)
Zhang, S., Choromanska, A.E., LeCun, Y.: Deep learning with elastic averaging SGD. In: Advances in Neural Information Processing Systems, pp. 685–693 (2015)
Acknowledgements
J.U. Rahman\(^{1}\), supported by CAS-TWAS President’s Fellowship at University of Science and Technology of China, No. 96, JinZhai Road Baohe District, Hefei, Anhui, 230026, P.R.China. We would also like to thanks Mr. Muhammad Ajmal from University of science and technology of China (USTC) for his valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Ul Rahman, J., Ali, A., Ur Rehman, M., Kazmi, R. (2020). A Unit Softmax with Laplacian Smoothing Stochastic Gradient Descent for Deep Convolutional Neural Networks. In: Bajwa, I., Sibalija, T., Jawawi, D. (eds) Intelligent Technologies and Applications. INTAP 2019. Communications in Computer and Information Science, vol 1198. Springer, Singapore. https://doi.org/10.1007/978-981-15-5232-8_14
Download citation
DOI: https://doi.org/10.1007/978-981-15-5232-8_14
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-5231-1
Online ISBN: 978-981-15-5232-8
eBook Packages: Computer ScienceComputer Science (R0)