# American Institute of Mathematical Sciences

February  2020, 3(1): 51-64. doi: 10.3934/mfc.2020005

## An improved deep convolutional neural network model with kernel loss function in image classification

 1 Key Laboratory of Education Informatization for Nationalities, Ministry of Education, Yunnan Normal University, Kunming 650500, China 2 School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China

* Corresponding author: Tianwei Xu

Received  December 2019 Revised  December 2019 Published  February 2020

Fund Project: This work is supported by National Natural Science Foundation of China (No. 61862068)

To further enhance the performance of the current convolutional neural network, an improved deep convolutional neural network model is shown in this paper. Different from the traditional network structure, in our proposed method the pooling layer is replaced by two continuous convolutional layers with $3 \times 3$ convolution kernel between which a dropout layer is added to reduce overfitting, and cross entropy kernel is used as loss function. Experimental results on Mnist and Cifar-10 data sets for image classification show that, compared to several classical neural networks such as Alexnet, VGGNet and GoogleNet, the improved network achieve better performance in learning efficiency and recognition accuracy at relatively shallow network depths.

Citation: Yuantian Xia, Juxiang Zhou, Tianwei Xu, Wei Gao. An improved deep convolutional neural network model with kernel loss function in image classification. Mathematical Foundations of Computing, 2020, 3 (1) : 51-64. doi: 10.3934/mfc.2020005
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##### References:
 [1] M. Abadi, A. Agarwal and P. Barham, et al., TensorFlow: Large-scale machine learning on heterogeneous distributed systems[J], arXiv preprint, arXiv: 1603.04467, 2016. [2] Z. An, S. Li and J. Wang, Generalization of deep neural network for bearing fault diagnosis under different working conditions using multiple kernel method,, Neurocomputing, 352 (2019), 42-53.  doi: 10.1016/j.neucom.2019.04.010. [3] V. Badrinarayanan, A. Kendall and R. Cipolla, SegNet: A deep convolutional encoder-decoder architecture for image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 39 (2017), 2481-2495.  doi: 10.1109/TPAMI.2016.2644615. [4] C. K. Chui, S. B. Lin and D. X. Zhou, Deep neural networks for rotation-invariance approximation and learning, Anal. Appl., 17 (2019), 737-772.  doi: 10.1142/S0219530519400074. [5] M. Courbariaux, I. Hubara and D. Soudry, et al., Binarized neural networks: Training deep neural networks with weights and activations constrained to +1 or -1[J], arXiv preprint, arXiv: 1602.02830, 2016. [6] K. He, X. Zhang and S. Ren, et al., Delving deep into rectifiers: Surpassing human-level performance on imaget classification, Proceeding of the 2015 IEEE International Conference on Computer Vision, Santiago, Chile, Publisher: IEEE, 2015, 1026–1034. doi: 10.1109/ICCV.2015.123. [7] K. He, X. Zhang, S. Ren, et al., Deep residual learning for image recognition, Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition. Las Vegas, NV, USA, Publisher: IEEE, 2016,770–778. doi: 10.1109/CVPR.2016.90. [8] M. Heydari, E. Shivanian and B. Azarnavid, An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order, J. Comput. Appl. Math., 361 (2019), 97-112.  doi: 10.1016/j.cam.2019.04.017. [9] G. Huang, Z. Liu and L. V. D. Maaten, et al., Densely connected convolutional networks, IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA, Publisher: IEEE, 2017, 2261–2269. doi: 10.1109/CVPR.2017.243. [10] Y. Jiang, Z. Liang and H. Gao, An improved constraint-based Bayesian network learning method using Gaussian kernel probability density estimator, Expert Syst. Appl., 113 (2018), 544-554.  doi: 10.1016/j.eswa.2018.06.058. [11] A. Krizhevsky, I. Sutskever and G. Hinton, ImageNet classification with deep convolutional neural networks, Commun. ACM, 60 (2017), 84-90.  doi: 10.1145/3065386. [12] Y. Lecun, L. Bottou and Y. Bengio, Gradient-based learning applied to document recognition,, Proceedings of the IEEE, 86 (1998), 2278-2324.  doi: 10.1109/5.726791. [13] Y. Lecun, Y. Bengio and G. Hinton, Deep learning, Nature, 521 (2015), 436-444. [14] Y. Lei and D. X. Zhou, Convergence of online mirror descent, Appl. Comput. Harmon. Anal., 48 (2020), 343-373.  doi: 10.1016/j.acha.2018.05.005. [15] S. B. Lin and D. X. Zhou, Optimal learning rates for kernel partial least squares, J. Fourier Anal. Appl., 24 (2018), 908-933.  doi: 10.1007/s00041-017-9544-8. [16] J. Niu, L. X. Sun and M. Q. Xu, et al., A reproducing kernel method for solving heat conduction equations with delay,, Appl. Math. Lett., 100 (2020), 106036, 7 pp. doi: 10.1016/j.aml.2019.106036. [17] Y. Qu, L. Lin and P. Shen, Joint hierarchical category structure learning and large-scale image classification, IEEE Trans. Image Process., 26 (2017), 4331-4346.  doi: 10.1109/TIP.2016.2615423. [18] A. Radford, L. Metz and S. Chintala, Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks[J], arXiv preprint, arXiv: 1511.06434, 2015. [19] K. Roushangar and S. Shahnazi, Bed load prediction in gravel-bed rivers using wavelet kernel extreme learning machine and meta-heuristic methods, Int. J. Environ. Sci. Te., 16 (2019), 8197-8208.  doi: 10.1007/s13762-019-02287-6. [20] K. Simonyan and A. Zisserman, Very Deep Convolutional Networks for Large-Scale Image Recognition[J]., arXiv preprint, arXiv: 1409.1556, 2015. [21] N. Srivastava, G. Hinton and A. Krizhevsky, Dropout: A simple way to prevent neural networks from overfitting, J. Mach. Learn. Res., 15 (2014), 1929-1958. [22] C. Szegedy, W. Liu and Y. Jia, et al., Going deeper with convolutions, Proceedings of the 2015 IEEE Conference on Computer Vision and Pattern Recognition, Boston, MA, USA, Publisher: IEEE, 2015. doi: 10.1109/CVPR.2015.7298594. [23] D. X. Zhou, Universality of deep convolutional neural networks, Appl. Comput. Harmon. Anal., 48 (2020), 787-794.  doi: 10.1016/j.acha.2019.06.004. [24] D. X. Zhou, Theory of deep convolutional neural networks: Downsampling, Neural Networks, 124 (2020), 319-327.  doi: 10.1016/j.neunet.2020.01.018.
Mini-network replacing the $3 \times 3$ convolutions
kernel size: $3 \times 3$, stride: 2
Max pooling operation, kernel size: $4\times 4$, stride: 2
Dropout workflow
The improved network structure
The curve of recognition accuracy of Alexnet network and improved network with the training times on cifar-10
The curve of recognition accuracy of VGGNet and improved network with the training times on cifar-10
The curve of recognition accuracy of Google network and improved network with the training times on cifar-10
The curve of recognition accuracy of Alexnet network and improved network with the training times on Minist
The curve of recognition accuracy of VGGNet and improved network with the training times on Minist
The curve of recognition accuracy of GoogleNet network and improved network with the training times on Mnist
 Parameters Value CPU: Intel core i9-9900k GPU: NVIDIA GeForce RTX 2080ti RAM: 16.0 GB OS: WIN10 64-bit Develop software: Python3.7 + TensorFlow framework (GPU mode)
 Parameters Value CPU: Intel core i9-9900k GPU: NVIDIA GeForce RTX 2080ti RAM: 16.0 GB OS: WIN10 64-bit Develop software: Python3.7 + TensorFlow framework (GPU mode)
 Network Cifar Mnist Alexnet: train acc:0.95, test acc:0.78 train acc:0.98, test acc:0.97 VGGNet: train acc:0.98, test acc:0.83 train acc:0.99, test acc:0.98 Google network: train acc:1.0, test acc:0.90 train acc:1.0, test acc:1.0 Improve network: train acc:1.0, test acc:0.94 train acc:1.0, test acc:1.0
 Network Cifar Mnist Alexnet: train acc:0.95, test acc:0.78 train acc:0.98, test acc:0.97 VGGNet: train acc:0.98, test acc:0.83 train acc:0.99, test acc:0.98 Google network: train acc:1.0, test acc:0.90 train acc:1.0, test acc:1.0 Improve network: train acc:1.0, test acc:0.94 train acc:1.0, test acc:1.0
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