Semi-supervised wavelet shrinkage

To estimate a possibly multivariate regression function gg under the general regression setup, y=g+ϵy=g+ϵ, one can use wavelet thresholding as an alternative to conventional nonparametric regression methods. Wavelet thresholding is a simple operation in the wavelet domain that selects a subset of coefficients corresponding to an estimator of gg when back-transformed. We propose an enhancement to wavelet thresholding by selecting a subset in a semi-supervised fashion in which the neighboring structure and classification function appropriate for wavelet domains are utilized. Wavelet coefficients are classified into two types: labeled, which have either strong or weak magnitudes, and unlabeled, which have in-between magnitudes. Both are connected to neighboring coefficients and belong to a low-dimensional manifold within the set of all wavelet coefficients. The decision to include a coefficient in the model depends not only on its magnitude but also on the labeled and the unlabeled coefficients from its neighborhood. We discuss the theoretical properties of the method and demonstrate its performance in simulated examples.