On the Approximation of Functional Classes Equipped with a Uniform Measure Using Ridge Functions

Maiorov, Vitaly; Meir, Ron

doi:10.1006/jath.1998.3305

Cited by 25 publications

(23 citation statements)

References 14 publications

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“…Without going into the details, as they are not relevant in what follows, they prove that for all functions in this set one can approximate in the¸norm from M L to within approximation error c n\P B\ , where c is some constant independent of n. It is also proven that for each n there exists a function in the set for which one cannot approximate from M L with approximation error less than c n\P B\ for some other constant c independent of n. In [14] it is shown that the set of functions for which this lower bound holds is of large measure. See [13,14] for details. The point we wish to make here is twofold.…”

Section: Introductionmentioning

confidence: 97%

Lower bounds for approximation by MLP neural networks

1999

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Section: Introductionmentioning

confidence: 97%

Lower bounds for approximation by MLP neural networks

1999

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“…In addition, Maiorov, Meir and Ratsaby [9] show that the set of functions for which the estimate n −r/(d−1) holds is of large measure. In other words, this is not simply a worst case setting.…”

Section: Introduction and The Main Resultsmentioning

confidence: 97%

Approximation of Sobolev classes by polynomials and ridge functions

Konovalov

Leviatan

Maiorov

2009

Journal of Approximation Theory

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Let W r p (B d ) be the usual Sobolev class of functions on the unit ball B d in R d , and W •,r p (B d ) be the subclass of all radial functions in W r p (B d ). We show that for the classes W •,r p (B d ) and W r p (B d ), the orders of best approximation by polynomials in L q (B d ) coincide. We also obtain exact orders of best approximation in L 2 (B d ) of the classes W •,r p (B d ) by ridge functions and, as an immediate consequence, we obtain the same orders in L 2 (B d ) for the usual Sobolev classes W r p (B d ).

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“…Maiorov [20] calculated this quantity and determined the degree of approximation for the nonlinear manifold of ridge functions which include the manifold of functions represented by artificial neural networks with one hidden layer. Maiorov, Meir, and Ratsaby [21], extended his result to the degree of approximation measured by a probabilistic (n, δ)-width with respect to a uniform measure over the target class and determined finite sample complexity bounds for model selection using neural networks [29]. For more works concerning probabilistic widths of classes see Traub et al [34], Maiorov and Wasilkowski [22].…”

Section: Lemma 1 (Uniform Sln For the Indicator Function Class) Let mentioning

confidence: 95%

On the Value of Partial Information for Learning from Examples

Maiorov

1997

Journal of Complexity

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The PAC model of learning and its extension to real valued function classes provides a well-accepted theoretical framework for representing the problem of learning a target function g(x) using a random sample {(Based on the uniform strong law of large numbers the PAC model establishes the sample complexity, i.e., the sample size m which is sufficient for accurately estimating the target function to within high confidence. Often, in addition to a random sample, some form of prior knowledge is available about the target. It is intuitive that increasing the amount of information should have the same effect on the error as increasing the sample size. But quantitatively how does the rate of error with respect to increasing information compare to the rate of error with increasing sample size? To answer this we consider a new approach based on a combination of information-based complexity of Traub et al. and Vapnik-Chervonenkis (VC) theory. In contrast to VC-theory where function classes of finite pseudo-dimension are used only for statistical-based estimation, we let such classes play a dual role of functional estimation as well as approximation. This is captured in a newly introduced quantity, ρ d (F ), which represents a nonlinear width of a function class F. We then extend the notion of the nth minimal radius of information and define a quantity I n, d (F ) which measures the minimal approximation error of the worst-case target g ∈ F by the family of function classes having pseudo-dimension d given partial information on g consisting of values taken by n linear operators. The error rates are calculated which leads to a quantitative notion of the value of partial information for the paradigm of learning from examples.

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On the Approximation of Functional Classes Equipped with a Uniform Measure Using Ridge Functions

Cited by 25 publications

References 14 publications

Lower bounds for approximation by MLP neural networks

Lower bounds for approximation by MLP neural networks

Approximation of Sobolev classes by polynomials and ridge functions

On the Value of Partial Information for Learning from Examples

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