Fetching the paper…
Reading the bibliography…
We propose a new approach for approximating functions in $C([0,1]^d)$ via Kolmogorov superposition theorem (KST) based on the linear spline interpolation of the outer function in the Kolmogorov representation.
Vitushkin, A.G.: On the hilbert’s thirteenth problem. Soviet Mathematics Doklady 95
1954
Earlier work this paper cites.
Kolmogorov, A.N.: On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Doklady Akademii Nauk 114
1957
Earlier work this paper cites.
Lorentz, G.G.: Metric entropy, widths, and superpositions of functions. The American Mathematical Monthly 69
1962
Earlier work this paper cites.
Sprecher, D.A.: Ph.d. dissertation. PhD thesis, University of Maryland (1963)
1963
Earlier work this paper cites.
Henkin, G.M.: Linear superpositions of continuously differentiable functions. Doklady Akademii Nauk 157
1964
Earlier work this paper cites.
Vitushkin, A.G.: A proof of the existence of analytic functions of several variables not representable by linear superpositions of continuously differentiable functions of fewer variables. Doklady Akademii Nauk 156
1964
Earlier work this paper cites.
Sprecher, D.A.: A representation theorem for continuous functions of several variables. Proceedings of the American Mathematical Society 16
1965
Earlier work this paper cites.
Sprecher, D.A.: On the structure of continuous functions of several variables. Transactions of the American Mathematical Society 115
1965
Earlier work this paper cites.
Ostrand, P.A.: Dimension of metric spaces and hilbert’s problem 13. Bulletin of the American Mathematical Society 71
1965
Earlier work this paper cites.
Lorentz, G.G.: Approximation of Functions. Selected Topics in Mathematics, (1966)
1966
Earlier work this paper cites.
Sprecher, D.A.: On the structure of representation of continuous functions of several variables as finite sums of continuous functions of one variable. Proceedings of the American Mathematical Society 17
1966
Earlier work this paper cites.
Fridman, B.: Improvement in the smoothness of functions in the kolmogorov superposition theorem. Doklady Akademii Nauk SSSR 177
1967
Earlier work this paper cites.
Sprecher, D.A.: An improvement in the superposition theorem of kolmogorov. Journal of Mathematical Analysis and Applications 38
1972
Earlier work this paper cites.
Doss, R.: A superposition theorem for unbounded continuous functions. Transactions of the American Mathematical Society 233
1977
Earlier work this paper cites.
Demko, S.: A superposition theorem for bounded continuous functions. Proceedings of the American Mathematical Society 66
1977
Earlier work this paper cites.
de Boor, C.: Efficient computer manipulation of tensor products. ACM Transactions on Mathematical Software 5
1979
Earlier work this paper cites.
Powell, M.J.: Approximation Theory and Methods. Cambridge University Press, New York (1981)
1981
Earlier work this paper cites.
Sternfeld, Y.: Dimension, superposition of functions, and separation of points, in compact metric spaces. Israel Journal of Mathematics 50
1985
Cited alongside, same era.
Hecht-Nielsen, R.: Kolmogorov’s mapping neural network existence theorem. In: Proceedings of the International Conference on Neural Networks, vol. 3, pp. 11–14. IEEE Press, New York (1987)
1987
Cited alongside, same era.
Girosi, F., Poggio, T.: Representation properties of networks: Kolmogorov’s theorem is irrelevant. Neural Computation 1
1989
Cited alongside, same era.
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Systems 2
1989
Cited alongside, same era.
Køurkovà, V.: Kolmogorov’s theorem is relevant. Neural Computation 3
1991
Cited alongside, same era.
Lai, M.-J., Schumaker, L.L.: Domain decomposition method for scattered data fitting. SIAM Journal on Numerical Analysis 47
2009
Later among the works it cites.
Feng, Z.: Hilbert’s 13th problem. PhD thesis, University of Pittsburgh (2010)
2010
Later among the works it cites.
Klusowski, J.M., Barron, A.R.: 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 64
2018
Later among the works it cites.
Guliyev, N.J., Ismailov, V.E.: Approximation capability of two hidden layer feedforward neural networks with fixed weights. Neurocomputing 316
2018
Later among the works it cites.
Montanelli, H., Yang, H.: Error bounds for deep relu networks using the kolmogorov–arnold superposition theorem. Neural Networks 129
2020
Later among the works it cites.
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
Køurkovà, V.: Kolmogorov’s theorem and multilayer neural networks. Neural Networks 5
1992
Cited alongside, same era.
Mhaskar, H., Micchelli, C.A.: Approximation by superposition of sigmoidal and radial basis functions. Advances in Applied Mathematics 13
1992
Cited alongside, same era.
Barron, A.R.: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Transactions on Information Theory 39
1993
Cited alongside, same era.
Lin, J.-N., Unbehauen, R.: On the realization of a kolmogorov network. Neural Computation 5
1993
Cited alongside, same era.
Barron, A.R.: Approximation and estimation bounds for artificial neural networks. Machine Learning 14
1994
Cited alongside, same era.
Pinkus, A.: Approximation theory of the MLP model in neural networks. Acta Numerica, 143–195 (1999)
1999
Cited alongside, same era.
Maiorov, V., Pinkus, A.: Lower bounds for approximation by mlp neural networks. Neurocomputing 25
1999
Cited alongside, same era.
2021
Later among the works it cites.
Morris, S.: Hilbert 13: Are there any genuine continuous multivariate real-valued functions? Bulletin of the American Mathematical Society 58
2021
Later among the works it cites.
Schmidt-Hieber, J.: The kolmogorov–arnold representation theorem revisited. Neural Networks 137
2021
Later among the works it cites.
DeVore, R., Hanin, B., Petrova, G.: Neural network approximation. Acta Numerica 30
2021
Later among the works it cites.
Siegel, J.W., Xu, J.: High-order approximation rates for shallow neural networks with cosine and ( R e L U ) k (ReLU)^{k} activation functions. Applied and Computational Harmonic Analysis 58
2022
Later among the works it cites.
Fakhoury, D., Fakhoury, E., Speleers, H.: Exsplinet: An interpretable and expressive spline-based neural network. Neural Networks 152
2022
Later among the works it cites.
Daubechies, I., DeVore, R., Foucart, S., Hanin, B., Petrova, G.: Nonlinear approximation and (deep) relu networks. Constructive Approximation 55
2022
Later among the works it cites.
Lai, M.-J., Lee, J.: A multivariate spline based collocation method for numerical solution of partial differential equations. SIAM J. Numerical Analysis 60
2022
Later among the works it cites.
2024
Closest in time.
Shen, Z.: Sparse solution technique in semi-supervised local clustering and high dimensional function approximation. PhD thesis, University of Georgia (2024)
2024
Closest in time.
Allen, K., Lai, M.-J., Shen, Z.: Maximal volume matrix cross approximation for image compression and least squares solution. Advances in Computational Mathematics 50
2024
Closest in time.