TY - JOUR

T1 - On the Approximation of Functional Classes Equipped with a Uniform Measure Using Ridge Functions

AU - Maiorov, Vitaly

AU - Meir, Ron

AU - Ratsaby, Joel

N1 - Funding Information:
* The work of V. Maiorov was partially supported by the center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel. The work of R. Meir was partially supported by a grant from the Israel Science Foundation. Support of the Ollendorff center at the Department of Electrical Engineering in the Technion is also acknowledged.

PY - 1999/7

Y1 - 1999/7

N2 - We introduce a construction of a uniform measure over a functional class Br which is similar to a Besov class with smoothness index r. We then consider the problem of approximating Br using a manifold Mn which consists of all linear manifolds spanned by n ridge functions, i.e., Mn={∑ni=1gi(a i·x):ai∈Sd-1, gi∈L2([-1, 1])}, x∈Bd. It is proved that for some subset A⊂Br of probabilistic measure 1-δ, for all f∈A the degree of approximation of Mn behaves asymptotically as 1/nr/(d-1). As a direct consequence the probabilistic (n, δ)-width for nonlinear approximation denoted as dn, δ(Br, μ, Mn), where μ is a uniform measure over Br, 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.

AB - We introduce a construction of a uniform measure over a functional class Br which is similar to a Besov class with smoothness index r. We then consider the problem of approximating Br using a manifold Mn which consists of all linear manifolds spanned by n ridge functions, i.e., Mn={∑ni=1gi(a i·x):ai∈Sd-1, gi∈L2([-1, 1])}, x∈Bd. It is proved that for some subset A⊂Br of probabilistic measure 1-δ, for all f∈A the degree of approximation of Mn behaves asymptotically as 1/nr/(d-1). As a direct consequence the probabilistic (n, δ)-width for nonlinear approximation denoted as dn, δ(Br, μ, Mn), where μ is a uniform measure over Br, 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.

UR - http://www.scopus.com/inward/record.url?scp=0013166467&partnerID=8YFLogxK

U2 - 10.1006/jath.1998.3305

DO - 10.1006/jath.1998.3305

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AN - SCOPUS:0013166467

SN - 0021-9045

VL - 99

SP - 95

EP - 111

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

IS - 1

ER -