Some New Approximation Theory Problems Arising from Learning Theory and Related Topics

Description

Approximation theory plays an important role in mathematical analysis of learningalgorithms and data analysis. This project aims at solving some new approximation theory problems arising from learning theory and related topics, and applying the solutions toanalyze some learning algorithms. We shall _rst consider the topic of learning with functional data by studying approximation abilities of functional linear regression algorithmsin a learning theory framework, where the covariance kernel will be analyzed by approximation theory and the approximation error of the regularization scheme with functionaldata in reproducing kernel Hilbert spaces will be estimated. We shall then estimate cover-ing numbers of a vector set involving 2-norm and 1-norm arising from one-bit compressedsensing and a function set arising from additive models in ultra-high dimensions withsequence p-norm penalties, and carry out error analysis for the corresponding learningalgorithms. A learning theory of one-bit compressed sensing will be developed, whichallows general measurement vectors. Confidence-based bounds will be provided for theexcess risk which together with a comparison relation yields bounds for the error norm ofthe approximation of sparse vectors. Finally, some other topics in compressed sensing willalso be considered by methods and ideas from learning theory and approximation theory.