Error bounds for deep ReLU networks using the Kolmogorov–Arnold superposition theorem

Hadrien Montanelli; Haizhao Yang

doi:10.1016/j.neunet.2019.12.013

Error bounds for deep ReLU networks using the Kolmogorov–Arnold superposition theorem

Research output: Contribution to journal › Article › peer-review

Abstract

We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov–Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.

Original language	English
Pages (from-to)	1-6
Number of pages	6
Journal	Neural Networks
Volume	129
DOIs	https://doi.org/10.1016/j.neunet.2019.12.013
Publication status	Published - 1 Sept 2020
Externally published	Yes

Keywords

Approximation theory
Curse of dimensionality
Deep ReLU networks
Kolmogorov–Arnold superposition theorem

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10.1016/j.neunet.2019.12.013

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title = "Error bounds for deep ReLU networks using the Kolmogorov–Arnold superposition theorem",

abstract = "We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov–Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.",

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author = "Hadrien Montanelli and Haizhao Yang",

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N2 - We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov–Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.

AB - We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov–Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.

KW - Approximation theory

KW - Curse of dimensionality

KW - Deep ReLU networks

KW - Kolmogorov–Arnold superposition theorem

U2 - 10.1016/j.neunet.2019.12.013

DO - 10.1016/j.neunet.2019.12.013

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VL - 129

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JO - Neural Networks

JF - Neural Networks

ER -

Error bounds for deep ReLU networks using the Kolmogorov–Arnold superposition theorem

Abstract

Keywords

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