Efficient approximation of high-dimensional functions with neural networks

K.-I. Funahashi, “On the approximate realization of continuous mappings by neural networks,” Neural Networks , vol. 2, no. 3, pp. 183–192, 1989. [Online]. Available: http://www.sciencedirect.com/science/article/pii/0893608089900038

1989

Earlier work this paper cites.

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Control Signals Systems , vol. 2, no. 4, pp. 303–314, 1989. [Online]. Available: https://doi.org/10.1007/BF02551274

1989

Earlier work this paper cites.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks , vol. 2, no. 5, pp. 359–366, 1989. [Online]. Available: http://www.sciencedirect.com/science/article/pii/0893608089900208

1989

Earlier work this paper cites.

K. Hornik, “Approximation capabilities of multilayer feedforward networks,” Neural Networks , vol. 4, no. 2, pp. 251–257, 1991. [Online]. Available: http://www.sciencedirect.com/science/article/pii/089360809190009T

1991

Earlier work this paper cites.

L. K. Jones, “A simple lemma on greedy approximation in Hilbert space and convergence rates for projection pursuit regression and neural network training,” Ann. Statist. , vol. 20, no. 1, pp. 608–613, 1992. [Online]. Available: https://doi.org/10.1214/aos/1176348546

1992

Earlier work this paper cites.

A. R. Barron, “Neural net approximation,” Proceedings of the 7th Yale Workshop on Adaptive and Learning Systems , pp. 69–72, 1992

1992

Earlier work this paper cites.

M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforward networks with a nonpolynomial activation function can approximate any function,” Neural Networks , vol. 6, no. 6, pp. 861–867, 1993. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0893608005801315

1993

Earlier work this paper cites.

A. R. Barron, “Universal approximation bounds for superpositions of a sigmoidal function,” IEEE Trans. Inform. Theory , vol. 39, no. 3, pp. 930–945, 1993. [Online]. Available: https://doi.org/10.1109/18.256500

1993

Earlier work this paper cites.

F. Girosi and G. Anzellotti, “Rates of convergence for radial basis functions and neural networks,” in Artificial Neural Networks for Speech and Vision , R. J. Mammone, Ed. Chapman & Hall, 1993, pp. 97–113

1993

Earlier work this paper cites.

H. N. Mhaskar, “Approximation properties of a multilayered feedforward artificial neural network,” Adv. Comput. Math. , vol. 1, no. 1, pp. 61–80, 1993. [Online]. Available: https://doi.org/10.1007/BF02070821

1993

Earlier work this paper cites.

——, “Approximation and estimation bounds for artificial neural networks,” Machine Learning , vol. 14, no. 1, pp. 115–133, Jan 1994. [Online]. Available: https://doi.org/10.1007/BF00993164

1994

Earlier work this paper cites.

H. N. Mhaskar and C. A. Micchelli, “Dimension-independent bounds on the degree of approximation by neural networks,” IBM Journal of Research and Development , vol. 38, no. 3, pp. 277–284, May 1994

1994

Earlier work this paper cites.

M. J. Donahue, C. Darken, L. Gurvits, and E. Sontag, “Rates of convex approximation in non-hilbert spaces,” Constructive Approximation , vol. 13, no. 2, pp. 187–220, Jun 1997. [Online]. Available: https://doi.org/10.1007/BF02678464

1997

Earlier work this paper cites.

L. Gurvits and P. Koiran, “Approximation and learning of convex superpositions,” Journal of Computer and System Sciences , vol. 55, no. 1, pp. 161–170, 1997. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0022000097915063

1997

Earlier work this paper cites.

V. Kůrková, P. C. Kainen, and V. Kreinovich, “Estimates of the number of hidden units and variation with respect to half-spaces,” Neural Networks , vol. 10, no. 6, pp. 1061–1068, 1997. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0893608097000282

1997

Earlier work this paper cites.

A. Pinkus, “Approximation theory of the MLP model in neural networks,” Acta Numerica , vol. 8, pp. 143–195, 1999

1999

Cited alongside, same era.

V. Maiorov and A. Pinkus, “Lower bounds for approximation by MLP neural networks,” Neurocomputing , vol. 25, no. 1, pp. 81–91, 1999. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0925231298001118

1999

Cited alongside, same era.

V. Kůrková and M. Sanguineti, “Comparison of worst case errors in linear and neural network approximation,” IEEE Transactions on Information Theory , vol. 48, no. 1, pp. 264–275, Jan 2002

2002

Cited alongside, same era.

V. Kůrková and M. Sanguineti, “Geometric upper bounds on rates of variable-basis approximation,” IEEE Transactions on Information Theory , vol. 54, no. 12, pp. 5681–5688, Dec 2008

2008

Cited alongside, same era.

H. W. Lin, M. Tegmark, and D. Rolnick, “Why does deep and cheap learning work so well?” J. Stat. Phys. , vol. 168, no. 6, pp. 1223–1247, 2017. [Online]. Available: https://doi.org/10.1007/s10955-017-1836-5

Original

2017

Later among the works it cites.

N. J. Guliyev and V. E. Ismailov, “On the approximation by single hidden layer feedforward neural networks with fixed weights,” Neural Networks , vol. 98, pp. 296–304, 2018. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0893608017302927

2018

Later among the works it cites.

J. M. Klusowski and A. R. Barron, “Approximation by combinations of relu and squared relu ridge functions with ℓ 1 \ell^{1} and ℓ 0 \ell^{0} controls,” IEEE Transactions on Information Theory , vol. 64, no. 12, pp. 7649–7656, Dec 2018

2018

Later among the works it cites.

N. J. Guliyev and V. E. Ismailov, “Approximation capability of two hidden layer feedforward neural networks with fixed weights,” Neurocomputing , vol. 316, pp. 262–269, 2018. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0925231218309111