Combining thresholding rules: a new way to improve the performance of wavelet estimators

Abstract

International audienceIn this paper, we address the situation where we cannot differentiate wavelet-based threshold procedures because their sets of well-estimated functions (maxisets) are not nested. As a generic solution, we propose to proceed via a combination of these procedures in order to achieve new procedures which perform better in the sense that the involved maxisets contain the union of the previous ones. Throughout the paper we propose illuminating interpretations of the maxiset results and provide conditions to ensure that this combination generates larger maxisets. As an example, we propose to combine vertical- and horizontal-block thresholding procedures that are already known to perform well. We discuss the limitation of our method, and we check our theoretical results through numerical experiments

A New Wavelet Threshold Determination Method Considering Interscale Correlation in Signal Denoising

“…Cai and Zhou [15] proposed a data-driven threshold determination method based on SURE criterion. Autin et al [16] put forward a new idea by combining different threshold rules.…”

Block‐threshold‐adapted Estimators via a Maxiset Approach

“…Finally, we would like to remark that controlling the maxiset results uniformly in m is also of primary importance when considering general thresholding rules for which the maxiset may not be embedded for different values of m as explained by Autin et al . ().…”

“…This allows us to address (at least theoretically) the important problem of the choice of the best value for m. Indeed, in such a case, m only calibrates the rate of convergence; hence, the best choice of m will be the one that ensures the fastest reconstruction, that is, the smallest value of m considered. Finally, we would like to remark that controlling the maxiset results uniformly in m is also of primary importance when considering general thresholding rules for which the maxiset may not be embedded for different values of m as explained by Autin et al (2012). To present our maxiset results for the BT estimators within the horizontal block thresholding families, we propose to use a large collection of rates of convergence, which are .mv ";p / 4s=1C2s (with s > 0).…”

Maxiset Point of View for Signal Detection in Inverse Problems

“…Note that a very usual criticism concerns the situation where estimation methods have non nested maxisets. Autin et al (2012) discuss this important aspect of the maxiset approach explaining that, first, it is is somehow normal to find that some estimation methods are better in estimating some specific functions. In such a case, examining the 'form' of the maxiset will bring interesting information.…”