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The study of universal approximation of arbitrary functions $f: \mathcal{X} \to \mathcal{Y}$ by neural networks has a rich and thorough history dating back to Kolmogorov (1957).
On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition
Andrei Nikolaevich Kolmogorov · 1957
Earlier work this paper cites.
Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
Earlier work this paper cites.
On the approximate realization of continuous mappings by neural networks
Ken-Ichi Funahashi · 1989
Earlier work this paper cites.
Bayesian learning for neural networks , volume 118
Radford M Neal · 1990
Earlier work this paper cites.
Approximation capabilities of multilayer feedforward networks
Kurt Hornik · 1991
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Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems
Tianping Chen and Hong Chen · 1995
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Computation with infinite neural networks
Christopher KI Williams · 1998
Earlier work this paper cites.
Neural network approximation of continuous functionals and continuous functions on compactifications
Maxwell B Stinchcombe · 1999
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Autoregressive forecasting of some functional climatic variations
Philippe C Besse, Hervé Cardot, and David B Stephenson · 2000
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Theoretical properties of functional multi layer perceptrons
Fabrice Rossi, Brieuc Conan-Guez, and François Fleuret · 2002
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Functional data analysis
James O Ramsay · 2004
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Gaussian processes for machine learning
Matthias Seeger · 2004
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Continuous neural networks
Nicolas Le Roux and Yoshua Bengio · 2007
Cited alongside, same era.
Steps toward deep kernel methods from infinite neural networks
Tamir Hazan and Tommi Jaakkola · 2015
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Learning infinite-layer networks: beyond the kernel trick
Amir Globerson and Roi Livni · 2016
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Solving ill-posed inverse problems using iterative deep neural networks
Jonas Adler and Ozan Öktem · 2017
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The expressive power of neural networks: A view from the width
Zhou Lu, Hongming Pu, Feicheng Wang, Zhiqiang Hu, and Liwei Wang · 2017
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Marta Garnelo, Jonathan Schwarz, Dan Rosenbaum, Fabio Viola, Danilo J Rezende, SM Eslami, and Yee Whye Teh · 2018
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Youngmin Cho and Lawrence K Saul · 2011
Cited alongside, same era.
High-dimensional statistics: A non-asymptotic viewpoint , volume 48
Martin J Wainwright · 2019
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