Blockwise SVD with error in the operator and application to blind deconvolution
Vareschi, Thomas; Picard, Dominique; Hoffmann, Marc; Delattre, Sylvain (2012), Blockwise SVD with error in the operator and application to blind deconvolution, Electronic Journal of Statistics, 6, p. 2274-2308. http://dx.doi.org/10.1214/12-EJS745
Type
Article accepté pour publication ou publiéDate
2012Nom de la revue
Electronic Journal of StatisticsVolume
6Éditeur
Institute of Mathematical Statistics
Pages
2274-2308
Identifiant publication
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Afficher la notice complèteRésumé (EN)
We consider linear inverse problems in a nonparametric statistical framework. Both the signal and the operator are unknown and subject to error measurements. We establish minimax rates of convergence under squared error loss when the operator admits a blockwise singular value decomposition (blockwise SVD) and the smoothness of the signal is measured in a Sobolev sense. We construct a nonlinear procedure adapting simultaneously to the unknown smoothness of both the signal and the operator and achieving the optimal rate of convergence to within logarithmic terms. When the noise level in the operator is dominant, by taking full advantage of the blockwise SVD property, we demonstrate that the block SVD procedure outperforms classical methods based on Galerkin projection or nonlinear wavelet thresholding. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.Mots-clés
error in the operator; linear in- verse problems; nonparametric adaptive estimation; circular and spherical deconvolution; blockwise SVD; Blind deconvolutionPublications associées
Affichage des éléments liés par titre et auteur.
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Muzy, Jean-François; Delattre, Sylvain; Hoffmann, Marc; Bacry, Emmanuel (2013) Article accepté pour publication ou publié
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Muzy, Jean-François; Hoffmann, Marc; Delattre, Sylvain; Bacry, Emmanuel (2013) Article accepté pour publication ou publié
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Doumic, Marie; Hoffmann, Marc (2021) Document de travail / Working paper
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Delattre, Sylvain; Fournier, Nicolas; Hoffmann, Marc (2016) Article accepté pour publication ou publié
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Cohen, Alexander T.; Ay, Cihan; Hainaut, Philippe; Décousus, Hervé; Hoffmann, Ulrich; Gaine, Sean; Coppens, Michiel; da Silva, Pedro Marques; Jiménez, David; Amann-Vesti, Beatrice; Brüggenjürgen, Bernd; Lévy, Pierre; Bastida, Julio Lopez; Vicaut, Eric; Laeis, Petra; Fronk, Eva-Maria; Zierhut, Wolfgang; Malzer, Thomas; Bramlage, Peter; Agnelli, Giancarlo (2018) Article accepté pour publication ou publié