Linear Widths of Function Spaces Equipped with the Gaussian Measure

Maiorov, Vitaly

doi:10.1006/jath.1994.1035

Cited by 33 publications

(27 citation statements)

References 6 publications

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“…More generally, 1-dimensional diffusions are analyzed by Dereich [11,12], who determines the exact asymptotic behavior of the quantization numbers for r ≥ 1 under rather mild smoothness assumptions. The asymptotic behavior of the Kolmogorov widths is determined by Maiorov [32,33] for the Brownian motion.…”

Section: Application To Diffusion Processesmentioning

confidence: 99%

Infinite-Dimensional Quadrature and Approximation of Distributions

et al. 2008

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We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (Monte Carlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As Communicated by Arieh Iserles. 392 Found Comput Math (2009) 9: 391-429 auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.

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Section: Application To Diffusion Processesmentioning

confidence: 99%

Infinite-Dimensional Quadrature and Approximation of Distributions

et al. 2008

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“…From the construction of the class B r one can see that for any 0 $ 1 there exists a subset A # B r such that +(A)=1&$. Quantities similar to (1) were considered in [25,11,14] where + was taken to be a Gaussian or Wiener measure and the approximation was linear.…”

Section: Vostrecov and Kreinesmentioning

confidence: 99%

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

Maiorov¹,

Meir²

1999

Journal of Approximation Theory

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We introduce a construction of a uniform measure over a functional class B r which is similar to a Besov class with smoothness index r. We then consider the problem of approximating B r using a manifold M n which consists of all linear manifolds spanned by n ridge functions, i.e.,It is proved that for some subset A/B r of probabilistic measure 1&$, for all f # A the degree of approximation of M n behaves asymptotically as 1Ân rÂ(d&1) . As a direct consequence the probabilistic (n, $ )-width for nonlinear approximation denoted as d n, $ (B r , +, M n ), where + is a uniform measure over B r , is similarly bounded. The lower bound holds also for the specific case of approximation using a manifold of one hidden layer neural networks with n hidden units. Academic Press

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“…The approximation problem on spaces with Wiener measure in the L 2 -norm was investigated in the book of Traub et al (1988). Papers concerned with average n-widths in Banach spaces include Buslaev (1988), Heinrich (1990), Maiorov (1990Maiorov ( , 1993a, Maiorov and Wasilkowski (1996), Mathe (1990), Ritter (1990), Pietsch (1980), and Traub et al (1988). The approximation of the functionals on Banach space with measure can be found in Lee and Wasilkowski, (1986).…”

Section: Introductionmentioning

confidence: 98%

“…The asymptotic n-widths d n (C, L q , Ͷ) n Ϫ1/2 were calculated in Maiorov (1990) for the case 2 Յ q Ͻ ȍ, and in Maiorov (1993a) for the case q ϭ ȍ. In this work we prove that the same estimate is also true for the case 1 Յ q Ͻ 2.…”

Section: Introductionmentioning

confidence: 99%