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To help understand the underlying mechanisms of neural networks (NNs), several groups have, in recent years, studied the number of linear regions $\ell$ of piecewise linear functions generated by deep neural networks (DNN).
Deep neural network approximation theory
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Approximation by superpositions of a sigmoidal function
Cybenko, G. (1989) · 1989
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Multilayer feedforward networks are universal approximators
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Image coding based on a fractal theory of iterated contractive image transformations
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Fractal and wavelet image compression techniques
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Michaeli, T. and Irani, M. (2013) · 2013
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Barnsley, M. F. (2014) · 2014
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Manning, C., Surdeanu, M., Bauer, J., Finkel, J., Bethard, S., and McClosky, D. (2014) · 2014
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On the number of linear regions of deep neural networks
Montufar, G. F., Pascanu, R., Cho, K., and Bengio, Y. (2014) · 2014
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Piczak, K. J. (2015) · 2015
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Notes on the number of linear regions of deep neural networks
Montúfar, G. (2017) · 2017
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Why and when can deep-but not shallow-networks avoid the curse of dimensionality: a review
Poggio, T., Mhaskar, H., Rosasco, L., Miranda, B., and Liao, Q. (2017) · 2017
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Qi, C. R., Su, H., Mo, K., and Guibas, L. J. (2017) · 2017
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Bounding and counting linear regions of deep neural networks
Serra, T., Tjandraatmadja, C., and Ramalingam, S. (2017) · 2017
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Error bounds for approximations with deep relu networks
Yarotsky, D. (2017) · 2017
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The power of depth for feedforward neural networks
Eldan, R. and Shamir, O. (2016) · 2016
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Why deep neural networks for function approximation?
Liang, S. and Srikant, R. (2016) · 2016
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Understanding deep neural networks with rectified linear units
Arora, R., Basu, A., Mianjy, P., and Mukherjee, A. (2018) · 2018
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Relu deep neural networks and linear finite elements
He, J., L. L. X. J. and Zheng, C. (2018) · 2018
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Nonlinear approximation and (deep) relu networks
Daubechies, I., DeVore, R., Foucart, S., Hanin, B., and Petrova, G. (2019) · 2019
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