On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks

H. N. Mhaskar; J. Prestin

doi:10.1049/cp:19970728

On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks

H. N. Mhaskar^*, J. Prestin

^*Corresponding author for this work

Institute of Mathematics

California State University Los Angeles

Abstract

We describe a system of nodes in [-1, 1]^s, where s≥1 is an integer, with the following property. Suppose the values at these nodes of any target function f having a prescribed number r of continuous derivatives on [-1, 1]^s are known. Given any ε>0, we can construct a generalized translation network with O(ε^-s/r) number of principal elements that approximates f uniformly within ε. Within a constant multiple, this is the theoretically minimal number of principal elements required for this purpose. The system of nodes is independent of the target function. We also investigate the effect of small perturbations in this system of nodes on the degree of approximation.

Original language	English
Title of host publication	Fifth International Conference on Artificial Neural Networks
Publication date	1997
DOIs	https://doi.org/10.1049/cp:19970728
Publication status	Published - 1997

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title = "On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks",

abstract = "We describe a system of nodes in [-1, 1]s, where s≥1 is an integer, with the following property. Suppose the values at these nodes of any target function f having a prescribed number r of continuous derivatives on [-1, 1]s are known. Given any ε>0, we can construct a generalized translation network with O(ε-s/r) number of principal elements that approximates f uniformly within ε. Within a constant multiple, this is the theoretically minimal number of principal elements required for this purpose. The system of nodes is independent of the target function. We also investigate the effect of small perturbations in this system of nodes on the degree of approximation.",

author = "Mhaskar, {H. N.} and J. Prestin",

year = "1997",

doi = "10.1049/cp:19970728",

language = "English",

series = "Fifth International Conference on Artificial Neural Networks",

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On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks. / Mhaskar, H. N.; Prestin, J.
Fifth International Conference on Artificial Neural Networks. 1997. (Fifth International Conference on Artificial Neural Networks).

Research output: Chapters in Books/Reports/Conference Proceedings › Conference contribution › peer-review

TY - GEN

T1 - On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks

AU - Mhaskar, H. N.

AU - Prestin, J.

PY - 1997

Y1 - 1997

N2 - We describe a system of nodes in [-1, 1]s, where s≥1 is an integer, with the following property. Suppose the values at these nodes of any target function f having a prescribed number r of continuous derivatives on [-1, 1]s are known. Given any ε>0, we can construct a generalized translation network with O(ε-s/r) number of principal elements that approximates f uniformly within ε. Within a constant multiple, this is the theoretically minimal number of principal elements required for this purpose. The system of nodes is independent of the target function. We also investigate the effect of small perturbations in this system of nodes on the degree of approximation.

AB - We describe a system of nodes in [-1, 1]s, where s≥1 is an integer, with the following property. Suppose the values at these nodes of any target function f having a prescribed number r of continuous derivatives on [-1, 1]s are known. Given any ε>0, we can construct a generalized translation network with O(ε-s/r) number of principal elements that approximates f uniformly within ε. Within a constant multiple, this is the theoretically minimal number of principal elements required for this purpose. The system of nodes is independent of the target function. We also investigate the effect of small perturbations in this system of nodes on the degree of approximation.

U2 - 10.1049/cp:19970728

DO - 10.1049/cp:19970728

M3 - Conference contribution

T3 - Fifth International Conference on Artificial Neural Networks

BT - Fifth International Conference on Artificial Neural Networks

ER -

On a Choice of Sampling Nodes for Optimal Approximation of Smooth Functions by Generalized Translation Networks

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