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This paper is devoted to studying the optimal expressive power of ReLU deep neural networks (DNNs) and its application in approximation via the Kolmogorov Superposition Theorem.
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A. N. Kolmogorov · 1957
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Metric entropy, widths, and superpositions of functions
G. G. Lorentz · 1962
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Proof of the existence of analytic functions of several complex variables which are not representable by linear superpositions of continuously differentiable functions of fewer variables(analytic functions of many variables which cannot be represented by linear superposition of continuously differentiated functions)
A. Vitushkin · 1964
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D. Sprecher · 1965
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D. A. Sprecher · 1965
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Approximation of functions
G. G. Lorentz · 1966
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B. Fridman · 1967
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N. Sauer · 1972
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A combinatorial problem; stability and order for models and theories in infinitary languages
S. Shelah · 1972
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An improvement in the superposition theorem of kolmogorov
D. A. Sprecher · 1972
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Kolmogorov’s mapping neural network existence theorem
R. Hecht-Nielsen · 1987
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Approximation by superpositions of a sigmoidal function
G. Cybenko · 1989
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Representation properties of networks: Kolmogorov’s theorem is irrelevant
F. Girosi and T. Poggio · 1989
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Multilayer feedforward networks are universal approximators
K. Hornik, M. Stinchcombe, and H. White · 1989
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A simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training
L. K. Jones · 1992
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Universal approximation bounds for superpositions of a sigmoidal function
A. R. Barron · 1993
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Dimension and superposition of bounded continuous functions on locally compact, separable metric spaces
Y. Hattori · 1993
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Multilayer feedforward networks with a nonpolynomial activation function can approximate any function
M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken · 1993
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A universal mapping for kolmogorov’s superposition theorem
D. A. Sprecher · 1993
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Aspects of the numerical analysis of neural networks
S. Ellacott · 1994
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Almost linear vc dimension bounds for piecewise polynomial networks
P. Bartlett, V. Maiorov, and R. Meir · 1998
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Nonlinear approximation
R. A. DeVore · 1998
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Approximation theory of the mlp model in neural networks
A. Pinkus · 1999
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The finite element method for elliptic problems
P. G. Ciarlet · 2002
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Kolmogorov’s spline network
B. Igelnik and N. Parikh · 2003
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An application of Kolmogorov’s superposition theorem to function reconstruction in higher dimensions
J. Braun · 2009
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On a constructive proof of kolmogorov’s superposition theorem
J. Braun and M. Griebel · 2009
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Rectified linear units improve restricted boltzmann machines
V. Nair and G. E. Hinton · 2010
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On the number of linear regions of deep neural networks
G. F. Montufar, R. Pascanu, K. Cho, and Y. Bengio · 2014
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Kolmogorov superposition theorem and its applications
X. Liu · 2015
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Exponential relu dnn expression of holomorphic maps in high dimension
J. A. Opschoor, C. Schwab, and J. Zech · 2019
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Nonlinear approximation via compositions
Z. Shen, H. Yang, and S. Zhang · 2019
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Error bounds for approximations with deep relu neural networks in w s , p w^{s,p} norms
I. Gühring, G. Kutyniok, and P. Petersen · 2020
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Relu deep neural networks and linear finite elements
J. He, L. Li, J. Xu, and C. Zheng · 2020
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Error bounds for deep relu networks using the kolmogorov–arnold superposition theorem
H. Montanelli and H. Yang · 2020
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Deep relu networks and high-order finite element methods
J. A. Opschoor, P. C. Petersen, and C. Schwab · 2020
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M. Telgarsky · 2015
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Deep residual learning for image recognition
K. He, X. Zhang, S. Ren, and J. Sun · 2016
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Identity mappings in deep residual networks
K. He, X. Zhang, S. Ren, and J. Sun · 2016
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Deep vs. shallow networks: An approximation theory perspective
H. N. Mhaskar and T. Poggio · 2016
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Wide residual networks
S. Zagoruyko and N. Komodakis · 2016
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An algorithm for computing lipschitz inner functions in kolmogorov’s superposition theorem
J. Actor and M. G. Knepley · 2017
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Deep network approximation characterized by number of neurons
Z. Shen, H. Yang, and S. Zhang · 2020
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Approximation rates for neural networks with general activation functions
J. W. Siegel and J. Xu · 2020
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Finite neuron method and convergence analysis
J. Xu · 2020
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Kolmogorov width decay and poor approximators in machine learning: Shallow neural networks, random feature models and neural tangent kernels
W. E and S. Wojtowytsch · 2021
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M.-J. Lai and Z. Shen · 2021
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Deep network approximation for smooth functions
J. Lu, Z. Shen, H. Yang, and S. Zhang · 2021
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Deep relu networks overcome the curse of dimensionality for generalized bandlimited functions
H. Montanelli, H. Yang, and Q. Du · 2021
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Hilbert 13: Are there any genuine continuous multivariate real-valued functions?
S. Morris · 2021
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The kolmogorov–arnold representation theorem revisited
J. Schmidt-Hieber · 2021
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Power series expansion neural network
Q. Chen, W. Hao, and J. He · 2022
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Nonlinear approximation and (deep) relu networks
I. Daubechies, R. DeVore, S. Foucart, B. Hanin, and G. Petrova · 2022
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Relu deep neural networks from the hierarchical basis perspective
J. He, L. Li, and J. Xu · 2022
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Optimal approximation rate of relu networks in terms of width and depth
Z. Shen, H. Yang, and S. Zhang · 2022
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High-order approximation rates for shallow neural networks with cosine and reluk activation functions
J. W. Siegel and J. Xu · 2022
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Sharp bounds on the approximation rates, metric entropy, and n-widths of shallow neural networks
J. W. Siegel and J. Xu · 2022
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Exponential relu neural network approximation rates for point and edge singularities
C. Marcati, J. A. Opschoor, P. C. Petersen, and C. Schwab · 2023
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Sharp lower bounds on interpolation by deep relu neural networks at irregularly spaced data
J. W. Siegel · 2023
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