On nonlinear condensation principles with rates

Dickmeis, W.; Nessel, R. J.; Wickeren, E. van

doi:10.1007/bf01171483

Cited by 17 publications

(9 citation statements)

References 10 publications

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“…We compute an r-th difference of x n at the interval endpoint 1 to get the resonance condition (17) and (19) are fulfilled.…”

Section: Lemma 1 (Impossible Inverse Estimate)mentioning

confidence: 99%

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On Sharpness of Error Bounds for Univariate Approximation by Single Hidden Layer Feedforward Neural Networks

Goebbels

2020

Results Math

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A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions. Single hidden layer feedforward neural networks with one input node perform such operations. Errors of best approximation can be expressed using moduli of smoothness of the function to be approximated (i.e., to be learned). In this context, the quantitative extension of the uniform boundedness principle indeed allows to construct counterexamples that show approximation rates to be best possible. Approximation errors do not belong to the little-o class of given bounds. By choosing piecewise linear activation functions, the discussed problem becomes free knot spline approximation. Results of the present paper also hold for non-polynomial (and not piecewise defined) activation functions like inverse tangent. Based on Vapnik-Chervonenkis dimension, first results are shown for the logistic function.

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“…We compute an r-th difference of x n at the interval endpoint 1 to get the resonance condition (17) and (19) are fulfilled.…”

Section: Lemma 1 (Impossible Inverse Estimate)mentioning

confidence: 99%

“…The condition replaces sub-additivity. Another extension of the uniform boundedness principle to non-sub-linear functionals is proved in [18]. But this version of the theorem is stated for a family of error functionals with two parameters that has to fulfill a condition of quasi lower semi-continuity.…”

Section: A Uniform Boundedness Principle With Ratesmentioning

confidence: 99%

On Sharpness of Error Bounds for Univariate Approximation by Single Hidden Layer Feedforward Neural Networks

Goebbels

2020

Results Math

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“…This is done in Section 2 via a rather abstract theorem, given in terms of sublinear operators in Banach spaces. In fact, these considerations continue our previous investigations [8][9][10] on quantitative uniform boundedness and condensation principles. Accordingly, the method of proof of Theorem 2.1 essentially consists in appropriate quantitative extensions of the familiar gliding hump method.…”

Section: Irt*f] = (Gs(lrs If ] )mentioning

confidence: 85%

“…[8][9][10]) which essentially correspond to (2.4-7)o(2.12,13) (cf. Corollary 3.1), the present resonance Theorem2.1 additionally yields comparison results of type (2.14, 15).…”

Section: [T~li> O~(z~)it~g~i-it~l_ I-it~(f~-l)imentioning

confidence: 99%

A general approach to counterexamples in numerical analysis

1984

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Summary. The purpose of this paper is to indicate a unified approach to quantitative negative results in numerical analysis. This is done via a rather general theorem which in fact subsumes our previous quantitative uniform boundedness principles. The proof is based upon a gliding hump method. The general theorem is exemplarily applied to discuss the sharpness of various direct and inverse approximation results, known for the compound trapezoidal rule and for the approximate solution of the heat equation. The treatment outlines a program which may also be worked out for other procedures.

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“…The central assumption in [1] was that the operators are sublinear, i.e., fulfill (4.2) and (4.3). Further, in [1] the sequences of operators were analyzed for fixed t. In [10,11,18] conditions were discussed that allow results for more general sets T ⊂ R. For convergence almost everywhere a new approach was developed in [19] that extends the theorem of Banach and Steinhaus. All papers [10,11,18,19] have in common that they need the sublinearity of the involved operators.…”

Section: The Threshold Operator and Basic Propertiesmentioning

confidence: 99%

On the Behavior of the Threshold Operator for Bandlimited Functions

Boche

Monich

2013

J Fourier Anal Appl

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One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley-Wiener space PW 1 π is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in PW 1 π , lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L ∞-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling. Math Subject Classifications. 94A20, 94A12.

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On nonlinear condensation principles with rates

Cited by 17 publications

References 10 publications

On Sharpness of Error Bounds for Univariate Approximation by Single Hidden Layer Feedforward Neural Networks

On Sharpness of Error Bounds for Univariate Approximation by Single Hidden Layer Feedforward Neural Networks

A general approach to counterexamples in numerical analysis

On the Behavior of the Threshold Operator for Bandlimited Functions

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