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In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded.
Threshold circuits of bounded depth
A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan · 1987
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Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
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On the approximate realization of continuous mappings by neural networks
K.-I. Funahashi · 1989
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Approximation capabilities of multilayer feedforward networks
K. Hornik · 1991
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Majority gates vs. general weighted threshold gates
M. Goldmann, J. Håstad, and A. Razborov · 1992
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On small depth threshold circuits
A. A. Razborov · 1992
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Neural computing with small weights
K.-Y. Siu and J. Bruck · 1992
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Approximation and estimation bounds for artificial neural networks
A. R. Barron · 1994
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On optimal depth threshold circuits for multiplication and related problems
K.-Y. Siu and V. P. Roychowdhury · 1994
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Bounds for the computational power and learning complexity of analog neural nets
W. Maass · 1997
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Natural proofs
A. A. Razborov and S. Rudich · 1997
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Simulating threshold circuits by majority circuits
M. Goldmann and M. Karpinski · 1998
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M. Krause and S. Lucks · 2001
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M. Naor and O. Reingold · 2004
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J. Martens, A. Chattopadhya, T. Pitassi, and R. Zemel · 2013
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M. Telgarsky · 2016
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Spectrally-normalized margin bounds for neural networks
P. L. Bartlett, D. J. Foster, and M. J. Telgarsky · 2017
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Depth separation for neural networks
A. Daniely · 2017
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Size-independent sample complexity of neural networks
N. Golowich, A. Rakhlin, and O. Shamir · 2017
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Exploring generalization in deep learning
B. Neyshabur, S. Bhojanapalli, D. McAllester, and N. Srebro · 2017
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M. Telgarsky · 2015
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The power of depth for feedforward neural networks
R. Eldan and O. Shamir · 2016
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Deep learning
I. Goodfellow, Y. Bengio, and A. Courville · 2016
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Why deep neural networks for function approximation?
S. Liang and R. Srikant · 2016
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Weight normalization: A simple reparameterization to accelerate training of deep neural networks
T. Salimans and D. P. Kingma · 2016
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Depth-width tradeoffs in approximating natural functions with neural networks
I. Safran and O. Shamir · 2017
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Error bounds for approximations with deep relu networks
D. Yarotsky · 2017
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Symmetric multivariate and related distributions
K. W. Fang · 2018
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Understanding weight normalized deep neural networks with rectified linear units
Y. Xu and X. Wang · 2018
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Depth separations in neural networks: What is actually being separated?
I. Safran, R. Eldan, and O. Shamir · 2019
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