Investigation Of Some Problems On Regularization Algorithms In Learning Theory

The main topic of this thesis is about spectral regularization algorithms in learning theory.Spectral regularization algorithms are based on reproducing kernel Hilbert spaces (RKHS).This class of algorithms aims at investigating the common natures of different regularizationalgorithms through regularization functions. Consistency analysis of spectral regularizationalgorithms and other related topics are fully discussed.The main method for discussing the performance of spectral regularization algorithms isthe technique of integral operator.The primary contributions of our work are the following two points: firstly, we generalizethe category of spectral regularization algorithms; secondly, we expand the regularizationconditions of existing literatures. For one thing, coefficient regularization algorithms withl2norm falls within the scope of spectral regularization algorithms. For another, regularizationconditions of previous literatures are restricted on RKHS, we extend it to the class of squareintegrable functions.Basing on the work above, we prove the consistency of spectral regularization algorithmsand deduce the error bound. Furthermore, we give the learning rates correspond to coefficientregularization withl2norm and least square situations. Simulation procedures are performedwhich aims at obtaining a direct-viewing understanding of some learning algorithms.Additionally, we discuss the consistency of spectral regularization algorithms integrating withsparsity conditions with respect to spectral regularization algorithms proved by literatures inexistence, and give the error bound.This thesis has also discussed other topics of learning theory: estimation of multivariatedistributions precision matrix learning and the property of asymmetric kernels. The primaryinnovation points of this section are: firstly, we turn the estimation of precision matrix into theestimation of covariance matrix; secondly, we investigate asymmetric kernels using the methodof one rank operators.