Wavelets and Approximation Theory Methods for Mathematical Analysis of Learning

Wavelet analysis is a well developed tool for time-frequency analysis by decomposing and analyzing data or functions according to different frequency components. Wavelets and related approximation theory methods can be used to extract and process information from massive data which is a main objective of learning theory. The purpose of this project is to study the interaction among learning theory, wavelet analysis and approximation theory. In particular, mathematical analysis of some learning schemes will be done by wavelets and approximation theory methods. Error analysis for the least square regression algorithm with ℓ1regularization penalty in sample dependent hypothesis spaces will be conducted. A moving least square learning method for regression will be analyzed by techniques from approximation theory of radial basis functions. A learning algorithm with sparse property for dimensionality reduction will be proposed and investigated by ideas from sparse wavelet methods using the ℓ1penalty. Dimensionality reduction and feature selection will be studied with discrete Laplacians of various orders by means of approximation of integral operators. Multi-task learning will be considered from a wavelet point of view. Some topics in wavelet analysis and approximation theory will also be investigated in probabilistic settings motivated by learning theory.