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Topological approaches to deep learning. (English) Zbl 1441.68220
Baas, Nils (ed.) et al., Topological data analysis. Proceedings of the Abel symposium 2018, Geiranger, Norway, June 4–8, 2018. Cham: Springer. Abel Symp. 15, 119-146 (2020).
Summary: In this work we introduce an algebraic formalism to describe and construct deep learning architectures as well as actions on them. We show how our algebraic formalism in conjunction with topological data analysis enables the construction of neural network architectures from a priori geometries, geometries obtained from data analysis, and purely data driven geometries. We also demonstrate how these techniques can improve the transparency and performance of deep neural networks.
For the entire collection see [Zbl 1448.62008].

MSC:
68T07 Artificial neural networks and deep learning
55N31 Persistent homology and applications, topological data analysis
Software:
CIFAR; ImageNet; MNIST
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References:
[1] G. Carlsson, Topology and data, Bull. Amer. Math. Soc. 46 (2009), 255-308 · Zbl 1172.62002
[2] G. Carlsson, T. Ishkhanov, V. de Silva, and A. Zomorodian, On the local behavior of spaces of natural images, Intl. Jour. Computer Vision, 76, (2008), 1-12
[3] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. FeiFei. Imagenet: A large-scale hierarchical image database, In IEEE Conference on Computer Vision and Pattern Recognition, 2009.
[4] R.B. Gabrielsson and G. Carlsson, A look at the topology of convolutional neural networks, arXiv:1810.03234v1
[5] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning, MIT Press, Adaptive computation and machine learning series, 2016. · Zbl 1373.68009
[6] L. Huang, A. Joseph, B. Nelson, B. Rubinstein, and J. Tygar, Adversarial machine learning, Proceedings of the 4th ACM workshop on security and artificial intelligence, ACM< 2011, 43-58, 2011.
[7] D. H. Hubel and T.N. Wiesel, Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex, Journal of Physiology, 160(1): 106-154, 1962.
[8] A. Krizhevsky, Learning multiple layers of features from tiny images, Technical report, University of Toronto, 2009.
[9] Y. LeCun, The MNIST database of handwritten digits, http://yann.lecun.com/exdb/mnist.
[10] A.B. Lee, K.S. Pedersen, and D. Mumford, The non-linear statistics of high-contrast patches in natural images, Intl. Jour.of Computer Vision, 54 (1-3), (2003) 83-103. · Zbl 1070.68661
[11] A. Maleki, M. Shahram, and G. Carlsson, Near optimal coder for image geometries, Proc. IEEE Int. Conf. Image Processing (ICIP), San Diego, CA, 2008 (paper can be found at https://www.ece.rice.edu/ mam15/Kleinlet_fullversion)
[12] K. Priddy and P. Keller, Artificial Neural Networks. An Introduction, SPIE Press, 2005.
[13] M.Robinson, Topological Signal Processing, Springer Verlag, 2014. · Zbl 1294.94001
[14] G. Singh, F. Mémoli, and G. Carlsson, Topological methods for the analysis of high dimensional data sets and 3D object recognition, SPBG, 2007, 91-100
[15] J. H. van Hateren and A. van der Schaaf, Independent component filters of natural images compared with simple cells in primary visual cortex, Proceedings of the Royal Society of London Series B, 265, 1998, 359-366.
[16] J. Perea and G. Carlsson, A Klein Bottle-based dictionary for texture representation, International Journal of Computer Vision, vol. 107, 75-97, 2014. · Zbl 1328.68279
[17] K. Simonyan and A. Zisserman, Very Deep Convolutional Networks for Large-Scale Image Recognition, CoRR 1409.1556, 2014.
[18] M. Tovée, An Introduction to the Visual System, Cambridge University Press, 2008
[19] http://ufldl.stanford.edu/housenumbers/
[20] Ayasdi, TDA and machine learning (2016) https://www.
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