טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentGeifman Maxim
SubjectHarmonic Analysis and Approximation by Ridge Functions
DepartmentDepartment of Applied Mathematics
Supervisors Professor Emeritus Allan Pinkus
Dr. Vitaly Maiorov


Abstract

We study approximation of multivariate functions by ridge functions. Ridge functions are defined as functions of the form h(a.x) , where a and x in Rd, (d>1),  h:R->R and a.x is the usual  inner product. In particular, we study approximation by sums of   n arbitrary ridge functions. Ridge functions or plane waves have many applications in various areas of mathematics, and in its applications. Different aspects of the approximation by ridge functions are natural components of many theoretical and applied problems, for instance, the Radon transform mathematical physics equations, geometry and tomography. Also, various approximation-theoretic problems that arise in the multilayer feedforward perceptron (MLP) model in neural networks are an important field of applications of ridge function theory. The question of approximation by ridge functions has been intensively investigated in recent years. It is known that the harmonic analysis of functions plays an important role in the theory of approximation. Our thesis is an attempt to study approximating properties of the ridge functions manifold by methods of harmonic analysis. Using the Radon transform we construct an orthogonal basis P of functions in the Hilbert space L2 on the unit ball. Each function is a linear combination of polynomial ridge functions. This basis is based on the construction of a convolution of two orthogonal bases in the Hilbert spaces on the unit sphere and on the segment [-1, 1], respectively. Next, we calculate Fourier coefficients (moments) of the special functions - ridge, radial and harmonic functions with regard to the basis P  and prove theorems of approximation of a different function class. In particular, we obtain new results in the multivariate case about the asymptotic behavior of the best approximation of radial and harmonic functions by ridges. We show that the best approximation of harmonic functions by the ridge manifold Rd  is much better than the best approximation by the polynomial space Pd,n. We consider greedy algorithms in ridge approximation. Such algorithms are known in mathematical statistics as projection pursuit regression and in signal processing as matching pursuit. They consist of step-wise, iterative constructions of linear combination of ridge functions. In addition we provide a practical greedy algorithm in the Hilbert space on the unit ball for approximating any function from this space with respect to the basis P.