Convergence of the Red-TOWER method for removing noise from data

By coupling the wavelet transform with a particular nonlinear shrinking function, the Red-telescopic optimal wavelet estimation of the risk (TOWER) method is introduced for removing noise from signals. It is shown that the method yields convergence of the 2 risk to the actual solution with optimal rate. Moreover, the method is proved to be asymptotically efficient when the regularization parameter is selected by the generalized cross validation criterion (GCV) or the Mallows criterion. Numerical experiments based on synthetic data are provided to compare the performance of the Red-TOWER method with hard-thresholding, soft-thresholding, and neigh–coeff thresholding. Furthermore, the numerical tests are also performed when the TOWER method is applied to hard-thresholding, soft-thresholding, and neigh–coeff thresholding, for which the full convergence results are still open.