Ridge Functions and Orthonormal Ridgelets

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets are a specialized set of angularly-integrated ridge functions which make up an orthonormal basis for L2(R). In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x1 cos θ + x2 sin θ). We derive a formula giving the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t), and w...

Ridgelets: Estimating with Ridge Functions

Feedforward neural networks, projection pursuit regression, and more generally, estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences. To address problems inherent to these methods – ranging from the construction of neural networks to their efficiency and capab...

Orthonormal Ridgelets and Linear Singularities

We construct a new orthonormal basis for L(R), whose elements are angularly integrated ridge functions — orthonormal ridgelets. The new basis functions are in L(R) and so are to be distinguished from the ridge function approximation system called ridgelets by Candès (1997, 1998), as ridge functions are not in L(R). Orthonormal ridgelet expansions have an interesting application in nonlinear app...

Rational Orthonormal Functions and Applications

Acknowledgement I am very grateful to Professor Ferenc Schipp for initiating me on the field of the approximation theory, for his advices and remarks without them this thesis would have been impossible. I am also indebted to the members of the Department of Numerical Analysis of the Eötvös Loránd University for their support during the years. I am especially grateful to Professor József Bokor i...

Ridgelets and the Representation of Mutilated Sobolev Functions