Unraveling Convolution Neural Networks: A Topological Exploration of Kernel Evolution
Source abstract
Abstract
Convolutional Neural Networks (CNNs) have become essential in deep learning applications, especially in computer vision, yet their complex internal mechanisms pose significant challenges to interpretability, crucial for ethical applications. Addressing this, our paper explores CNNs by examining their topological changes throughout the learning process, specifically employing persistent homology, a core method within Topological Data Analysis (TDA), to observe the dynamic evolution of their structure. This approach allows us to identify consistent patterns in the topological features of CNN kernels, particularly through shifts in Betti curves, which is a key concept in TDA. Our analysis of these Betti curves, initially focusing on the zeroth and first Betti numbers (respectively referred to as Betti-0 and Betti-1, which denote the number of connected components and loops), reveals insights into the learning dynamics of CNNs and potentially indicates the effectiveness of the learning process. We also discover notable differences in topological structures when CNNs are trained on grayscale versus color datasets, indicating the need for more extensive parameter space adjustments in color image processing. This study not only enhances the understanding of the intricate workings of CNNs but also contributes to bridging the gap between their complex operations and practical, interpretable applications.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Topological and Geometric Data Analysis · Cell Image Analysis Techniques · Model Reduction and Neural Networks
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
College of Science, Chongqing University of Technology, Chongqing 400054, China
Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China
Funding
Scientific Research Foundation of Chongqing University of Technology · Science and Technology Research Program of Chongqing Municipal Education Commission
References (36)
Show source-provided references
1. Backpropagation applied to handwritten zip code recognition10.1162/neco.1989.1.4.541 ↗
2. Imagenet classification with deep convolutional neural networks10.1145/3065386 ↗
3. Simonyan, K., and Zisserman, A. (2014). Very deep convolutional networks for large-scale image recognition. arXiv.
4. Understanding deep learning (still) requires rethinking generalization10.1145/3446776 ↗
5. Li, X., Chen, S., Hu, X., and Yang, J. (2019, January 15–19). Understanding the disharmony between dropout and batch normalization by variance shift. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA.10.1109/CVPR.2019.00279 ↗
6. Brundage, M., Avin, S., Clark, J., Toner, H., Eckersley, P., Garfinkel, B., Dafoe, A., Scharre, P., Zeitzoff, T., and Filar, B. (2018). The malicious use of artificial intelligence: Forecasting, prevention, and mitigation. arXiv.
7. Binns, R. (2018, January 23–24). Fairness in machine learning: Lessons from political philosophy. Proceedings of the Conference on Fairness, Accountability and Transparency, New York, NY, USA.
8. Ribeiro, M.T., Singh, S., and Guestrin, C. (2016, January 13–17). “Why should i trust you?” Explaining the predictions of any classifier. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA.10.1145/2939672.2939778 ↗
9. Lundberg, S.M., and Lee, S.I. (2017, January 4–9). A unified approach to interpreting model predictions. Proceedings of the Advances in Neural Information Processing Systems 30 (NIPS 2017), Long Beach, CA, USA.
10. Zeiler, M.D., and Fergus, R. (2014). Proceedings of the Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, 6–12 September 2014, Proceedings, Part I 13, Springer.
11. Keskar, N.S., Mudigere, D., Nocedal, J., Smelyanskiy, M., and Tang, P.T.P. (2016). On large-batch training for deep learning: Generalization gap and sharp minima. arXiv.
12. Bartlett, P.L., Foster, D.J., and Telgarsky, M.J. (2017, January 4–9). Spectrally-normalized margin bounds for neural networks. Proceedings of the Advances in Neural Information Processing Systems 30 (NIPS 2017), Long Beach, CA, USA.
13. Carlini, N., and Wagner, D. (2017, January 22–26). Towards evaluating the robustness of neural networks. Proceedings of the 2017 IEEE Symposium on Security and Privacy (SP), San Jose, CA, USA.10.1109/SP.2017.49 ↗
14. Gebhart, T., Schrater, P., and Hylton, A. (2019, January 16–19). Characterizing the shape of activation space in deep neural networks. Proceedings of the 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), Boca Raton, FL, USA.10.1109/ICMLA.2019.00254 ↗
15. Rieck, B., Togninalli, M., Bock, C., Moor, M., Horn, M., Gumbsch, T., and Borgwardt, K. (2018). Neural persistence: A complexity measure for deep neural networks using algebraic topology. arXiv.
16. Topological measurement of deep neural networks using persistent homology10.1007/s10472-021-09761-3 ↗
17. Topology of deep neural networks
18. A topological loss function for deep-learning based image segmentation using persistent homology10.1109/TPAMI.2020.3013679 ↗
19. Ballester, R., Casacuberta, C., and Escalera, S. (2023). Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey. arXiv.
20. A survey of vectorization methods in topological data analysis10.1109/TPAMI.2023.3308391 ↗
21. Persistence paths and signature features in topological data analysis10.1109/TPAMI.2018.2885516 ↗
22. On the complexity of neural network classifiers: A comparison between shallow and deep architectures10.1109/TNNLS.2013.2293637 ↗
23. Guss, W.H., and Salakhutdinov, R. (2018). On characterizing the capacity of neural networks using algebraic topology. arXiv.
24. Goldfarb, D. (2018). Understanding deep neural networks using topological data analysis. arXiv.
25. Gabrielsson, R.B., and Carlsson, G. (2019, January 16–19). Exposition and interpretation of the topology of neural networks. Proceedings of the 2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA), Boca Raton, FL, USA.
26. Carlsson, G., and Gabrielsson, R.B. (2018, January 4–8). Topological approaches to deep learning. Proceedings of the Topological Data Analysis: The Abel Symposium 2018, Geiranger, Norway.
27. Zomorodian, A., and Carlsson, G. (2004, January 8–11). Computing persistent homology. Proceedings of the Twentieth Annual Symposium on Computational Geometry, New York, NY, USA.10.1145/997817.997870 ↗
28. Persistent-homology-based machine learning: A survey and a comparative study10.1007/s10462-022-10146-z ↗
29. Carlsson, G., and Zomorodian, A. (2007, January 6–8). The theory of multidimensional persistence. Proceedings of the Twenty-Third Annual Symposium on Computational Geometry, Gyeongju, Republic of Korea.10.1145/1247069.1247105 ↗
30. Topological methods for the analysis of high dimensional data sets and 3d object recognition
31. LeCun, Y., Cortes, C., and Burges, C. (2023, October 04). The MNIST Database of Handwritten Digits. Available online: http://yann.lecun.com/exdb/mnist/.
32. Clanuwat, T., Bober-Irizar, M., Kitamoto, A., Lamb, A., Yamamoto, K., and Ha, D. (2023, October 04). Deep Learning for Classical Japanese Literature. Available online: https://github.com/rois-codh/kmnist.
33. Xiao, H., Rasul, K., and Vollgraf, R. (2023, October 04). Fashion-MNIST: A Novel Image Dataset for Benchmarking Machine Learning Algorithms. Available online: https://github.com/zalandoresearch/fashion-mnist.
34. Krizhevsky, A. (2023, October 04). Learning Multiple Layers of Features from Tiny Images. Available online: https://www.cs.toronto.edu/~kriz/cifar.html.
35. Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., and Antiga, L. (2019, January 8–14). Pytorch: An imperative style, high-performance deep learning library. Proceedings of the Advances in Neural Information Processing Systems 32 (NeurIPS 2019), Vancouver, BC, Canada.
36. Ripser: Efficient computation of Vietoris–Rips persistence barcodes10.1007/s41468-021-00071-5 ↗