Learning with trigonometric polynomials

Abstract

In this thesis, we will focus on two problems in learning theory. Firstly, we will discuss the problem of reconstruction of multivariable trigonometric polynomials. This is a special example of learning in finite dimensional spaces. To estimate the constant number of least square algorithm, we derive an inequality for the Hilbert-Schmidt norm of the difference between the sample second moment matrix m-1ULU and its expectation by a probability inequality of Hilbert space valued variables where U is a complex random m x D matrix with independent rows. This result immediately implies a bound on condition number of the sample second moment matrix. The results provide a solid theoretical foundation for those efficient numerical algorithms. A relaxed condition was used to generalize the result about trigonometric polynomials. Secondly, we will consider computing the empirical task function of regularity problems. This could be considered as an extension of the results of learning with trigonometric polynomials in our points of view. We will give a bound about the condition number of the regularity algorithm.