Refutation of a theorem of Diliberto and Straus

Medvedev, V. A.

doi:10.1007/bf01250549

Cited by 7 publications

(4 citation statements)

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“…This conjecture was first proposed in 1951 by Diliberto and Straus [5]. But later it was shown by Aumann [2], and independently by Medvedev [22], that the algorithm may not converge to E(f ) for the case n > 2.…”

Section: Consider the Iterationsmentioning

confidence: 98%

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Asgarova

Ismailov

2017

Proc Math Sci

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In 1951, Diliberto and Straus [5] proposed a levelling algorithm for the uniform approximation of a bivariate function, defined on a rectangle with sides parallel to the coordinate axes, by sums of univariate functions. In the current paper, we consider the problem of approximation of a continuous function defined on a compact Hausdorff space by a sum of two closed algebras containing constants. Under reasonable assumptions, we show the convergence of the Diliberto-Straus algorithm. For the approximation by sums of univariate functions, it follows that Diliberto-Straus's original result holds for a large class of compact convex sets.Mathematics Subject Classifications: 41A30, 41A65, 46B28, 65D15

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Section: Consider the Iterationsmentioning

confidence: 98%

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Asgarova

Ismailov

2017

Proc Math Sci

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“…Trying to generalize the notion of lightning bolt to this case we face big combinatorial-geometric difficulties. The generalizations in [2] and [8] have some gaps (see [6] and [16] on these gaps). This interesting question remains essentially open still.…”

Section: Remarkmentioning

confidence: 98%

Methods for computing the least deviation from the sums of functions of one variable

Ismailov

2006

Sib Math J

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“…There were several attempts to fill this gap in the special case when r = d and a 1 , ...,a r are the unit vectors. Unfortunately, all these attempts failed (see, for example, the attempts in [26,37] and the refutations in [4,28,109]).…”

Section: Density Of Ridge Functions and Some Problemsmentioning

confidence: 99%

Notes on ridge functions and neural networks

Ismailov¹

2020

Preprint

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To the Memory of My Parents PrefaceThese notes are about ridge functions. Recent years have witnessed a flurry of interest in these functions. Ridge functions appear in various fields and under various guises. They appear in fields as diverse as partial differential equations (where they are called plane waves), computerized tomography and statistics. These functions are also the underpinnings of many central models in neural networks.We are interested in ridge functions from the point of view of approximation theory. The basic goal in approximation theory is to approximate complicated objects by simpler objects. Among many classes of multivariate functions, linear combinations of ridge functions are a class of simpler functions. These notes study some problems of approximation of multivariate functions by linear combinations of ridge functions. We present here various properties of these functions. The questions we ask are as follows. When can a multivariate function be expressed as a linear combination of ridge functions from a certain class? When do such linear combinations represent each multivariate function? If a precise representation is not possible, can one approximate arbitrarily well? If well approximation fails, how can one compute/estimate the error of approximation, know that a best approximation exists? How can one characterize and construct best approximations? We also study properties of generalized ridge functions, which are very much related to linear superpositions and Kolmogorov's famous superposition theorem. These notes end with a few applications of ridge functions to the problem of approximation by single and two hidden layer neural networks with a restricted set of weights.We hope that these notes will be useful and interesting to both researchers and students.2 Approximation of multivariate functions by sums of univariate functions 2.1 Characterization of some bivariate function classes by formulas

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Refutation of a theorem of Diliberto and Straus

Cited by 7 publications

References 2 publications

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras

Methods for computing the least deviation from the sums of functions of one variable

Notes on ridge functions and neural networks

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