On error formulas for approximation by sums of univariate functions

Ismailov, Vugar E.

doi:10.1155/ijmms/2006/65620

Cited by 9 publications

(6 citation statements)

References 15 publications

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“…The idea of bolts was first introduced in Diliberto and Straus [26], where these objects are called "permissible lines". They appeared further in a number of papers, although under several different names (see, e.g., [18,29,34,36,55,56,60,62,76,79,82,93,107,108,113]). Note that the term "bolt of lightning" is due to Arnold [3].…”

Section: The Characterization Theoremmentioning

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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Section: The Characterization Theoremmentioning

confidence: 99%

Notes on ridge functions and neural networks

Ismailov¹

2020

Preprint

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“…Paths, in the special case when Q ⊂ R 2 , a 1 and a 2 coincide with the coordinate directions, are geometrically explicit objects. In this case, a path is a finite ordered set (p 1 , ..., p n ) in R 2 with the line segments [p i , p i+1 ], i = 1, ..., n, alternatively perpendicular to the x and y axes (see, e.g., [2,8,10,17,18,20,28]). These objects were first introduced by Diliberto and Straus [7] (in [7], they are called "permissible lines").…”

Section: Equioscillation Theorem For Ridge Functionsmentioning

confidence: 99%

A note on the equioscillation theorem for best ridge function approximation

Ismailov

2017

Expositiones Mathematicae

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Abstract. We consider the approximation of a continuous function, defined on a compact set of the d-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best approximation. The result resembles the classical Chebyshev equioscillation theorem for polynomial approximation.Mathematics Subject Classifications: 41A30, 41A50, 46B50, 46E15

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“…If a and b are the coordinate vectors in R 2 , then the objects in Definition 2.1 turn into "bolts of lightning" (see, e.g., [1,5,29]). It is well known that the idea of bolts was first introduced by Diliberto and Straus [8] and played an essential role in problems of approximation by sums of univariate functions (see, e.g., [8,11,14,22,23,28,29]). Note that the name "bolt of lightning" is due to Arnold [1].…”

Section: The Approximation Error Formulamentioning

confidence: 99%

“…We see that the approximation of a bivariate function by sums of univariate functions is a special case of the approximation problem considered in this paper. It should be remarked that there are many papers devoted to this subject (see, e.g., [2,8,11,14,22,23,28,29,36] and references therein).…”

Section: Introductionmentioning

confidence: 99%