Wavelet shrinkage is a strategy to obtain a nonlinear approximation to a given function f and is widely used in data compression,signal processing and statistics,etc.For Calder′on-Zygmund operators T,it is interesting to construct estimator of T f,based on wavelet shrinkage estimator of f.With the help of a representation of operators on wavelets,due to Beylkin et al.,an estimator of T f is presented in this paper.The almost everywhere convergence and norm convergence of the proposed estimators are established.
Semi-supervised learning is an emerging computational paradigm for machine learning,that aims to make better use of large amounts of inexpensive unlabeled data to improve the learning performance.While various methods have been proposed based on different intuitions,the crucial issue of generalization performance is still poorly understood.In this paper,we investigate the convergence property of the Laplacian regularized least squares regression,a semi-supervised learning algorithm based on manifold regularization.Moreover,the improvement of error bounds in terms of the number of labeled and unlabeled data is presented for the first time as far as we know.The convergence rate depends on the approximation property and the capacity of the reproducing kernel Hilbert space measured by covering numbers.Some new techniques are exploited for the analysis since an extra regularizer is introduced.
