NEED-VD: a second-generation wavelet algorithm for estimation in inverse problemsReport as inadecuate

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1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires 2 Department of Mathematics Columbia

Abstract : We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames termed ``needlets- built upon the SVD bases. We consider two different situations : the ``wavelet- scenario, where the needlets are assumed to behave similarly to true wavelets, and the ``Jacobi-type- scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand hence on the operator. To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L 2$ case, we obtain interesting rates of convergence for other $L p$ norms which are new to the best of our knowledge in the literature, and we also give a simulation study showing that the NEED-VD estimator outperforms other standard algorithms in almost all situations.

Keywords : statistical inverse problems minimax estimation second-generation wavelets

Author: Gérard Kerkyacharian - Pencho Petrushev - Dominique Picard - Thomas Willer -



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