Electronic Journal of Statistics

Blockwise SVD with error in the operator and application to blind deconvolution

S. Delattre, M. Hoffmann, D. Picard, and T. Vareschi

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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 [14] or nonlinear wavelet thresholding [18]. We subsequently apply our abstract framework to the specific case of blind deconvolution on the torus and on the sphere.

Article information

Electron. J. Statist. Volume 6 (2012), 2274-2308.

First available in Project Euclid: 30 November 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G99: None of the above, but in this section
Secondary: 65J20: Improperly posed problems; regularization 65J22: Inverse problems

Blind deconvolution blockwise SVD circular and spherical deconvolution nonparametric adaptive estimation linear inverse problems error in the operator


Delattre, S.; Hoffmann, M.; Picard, D.; Vareschi, T. Blockwise SVD with error in the operator and application to blind deconvolution. Electron. J. Statist. 6 (2012), 2274--2308. doi:10.1214/12-EJS745. http://projecteuclid.org/euclid.ejs/1354284420.

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