Volume 39, Issue 6
Deep ReLU Networks Overcome the Curse of Dimensionality for Generalized Bandlimited Functions

Hadrien Montanelli, Haizhao Yang & Qiang Du

J. Comp. Math., 39 (2021), pp. 801-815.

Published online: 2021-10

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  • Abstract

We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

  • Keywords

Machine learning, Deep ReLU networks, Curse of dimensionality, Approximation theory, Bandlimited functions, Chebyshev polynomials.

  • AMS Subject Headings

68T01, 33F05, 41A10

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

hadrien.montanelli@polytechnique.edu (Hadrien Montanelli)

haizhao@purdue.edu (Haizhao Yang)

qd2125@columbia.edu (Qiang Du)

  • BibTex
  • RIS
  • TXT
@Article{JCM-39-801, author = {Montanelli , Hadrien and Yang , Haizhao and Du , Qiang}, title = {Deep ReLU Networks Overcome the Curse of Dimensionality for Generalized Bandlimited Functions}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {39}, number = {6}, pages = {801--815}, abstract = {

We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2007-m2019-0239}, url = {http://global-sci.org/intro/article_detail/jcm/19912.html} }
TY - JOUR T1 - Deep ReLU Networks Overcome the Curse of Dimensionality for Generalized Bandlimited Functions AU - Montanelli , Hadrien AU - Yang , Haizhao AU - Du , Qiang JO - Journal of Computational Mathematics VL - 6 SP - 801 EP - 815 PY - 2021 DA - 2021/10 SN - 39 DO - http://doi.org/10.4208/jcm.2007-m2019-0239 UR - https://global-sci.org/intro/article_detail/jcm/19912.html KW - Machine learning, Deep ReLU networks, Curse of dimensionality, Approximation theory, Bandlimited functions, Chebyshev polynomials. AB -

We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

Hadrien Montanelli, Haizhao Yang & Qiang Du. (2021). Deep ReLU Networks Overcome the Curse of Dimensionality for Generalized Bandlimited Functions. Journal of Computational Mathematics. 39 (6). 801-815. doi:10.4208/jcm.2007-m2019-0239
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