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Spatially inhomogeneous linear inverse problems with possible singularities. (English) Zbl 1293.62014
Summary: The objective of the present paper is to introduce the concept of a spatially inhomogeneous linear inverse problem where the degree of ill-posedness of operator \(Q\) depends not only on the scale but also on location. In this case, the rates of convergence are determined by the interaction of four parameters, the smoothness and spatial homogeneity of the unknown function \(f\) and degrees of ill-posedness and spatial inhomogeneity of operator \(Q\).
Estimators obtained in the paper are based either on wavelet-vaguelette decomposition (if the norms of all vaguelettes are finite) or on a hybrid of wavelet-vaguelette decomposition and Galerkin method (if vaguelettes in the neighborhood of the singularity point have infinite norms). The hybrid estimator is a combination of a linear part in the vicinity of the singularity point and the nonlinear block thresholding wavelet estimator elsewhere. To attain adaptivity, an optimal resolution level for the linear, singularity affected, portion of the estimator is obtained using O. V. Lepskij [Theory Probab. Appl. 36, No. 4, 682–697 (1991; Zbl 0776.62039)] method and is used subsequently as the lowest resolution level for the nonlinear wavelet estimator. We show that convergence rates of the hybrid estimator lie within a logarithmic factor of the optimal minimax convergence rates.
The theory presented in the paper is supplemented by examples of deconvolution with a spatially inhomogeneous kernel and deconvolution in the presence of locally extreme noise or extremely inhomogeneous design. The first two problems are examined via a limited simulation study which demonstrates advantages of the hybrid estimator when the degree of spatial inhomogeneity is high. In addition, we apply the technique to recovery of a convolution signal transmitted via amplitude modulation.

62C10 Bayesian problems; characterization of Bayes procedures
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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[1] Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115-129. · Zbl 0908.62095 · doi:10.1093/biomet/85.1.115 · www3.oup.co.uk
[2] Antoniadis, A., Pensky, M. and Sapatinas, T. (2013). Nonparametric regression estimation based on spatially inhomogeneous data: Minimax global convergence rates and adaptivity. ESAIM Probab. Stat. · Zbl 1305.62167
[3] Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F. (2007). Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 2610-2636. · Zbl 1234.62062 · doi:10.1137/060651884
[4] Cavalier, L. and Golubev, Y. (2006). Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 1653-1677. · Zbl 1246.62082 · doi:10.1214/009053606000000542
[5] Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843-874. · Zbl 1029.62032 · doi:10.1214/aos/1028674843 · euclid:aos/1028674843
[6] Cohen, A., Hoffmann, M. and Reiß, M. (2004). Adaptive wavelet Galerkin methods for linear inverse problems. SIAM J. Numer. Anal. 42 1479-1501 (electronic). · Zbl 1077.65054 · doi:10.1137/S0036142902411793
[7] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101-126. · Zbl 0826.65117 · doi:10.1006/acha.1995.1008
[8] Engl, H. W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems . Kluwer Academic, Dordrecht. · Zbl 0859.65054
[9] Gaïffas, S. (2005). Convergence rates for pointwise curve estimation with a degenerate design. Math. Methods Statist. 14 1-27.
[10] Gaïffas, S. (2007a). Sharp estimation in sup norm with random design. Statist. Probab. Lett. 77 782-794. · Zbl 1114.62046 · doi:10.1016/j.spl.2006.11.017
[11] Gaïffas, S. (2007b). On pointwise adaptive curve estimation based on inhomogeneous data. ESAIM Probab. Stat. 11 344-364 (electronic). · Zbl 1187.62074 · doi:10.1051/ps:2007023 · numdam:PS_2007__11__344_0 · eudml:104378
[12] Gaïffas, S. (2009). Uniform estimation of a signal based on inhomogeneous data. Statist. Sinica 19 427-447. · Zbl 1168.62036 · www3.stat.sinica.edu.tw
[13] Golubev, Y. (2010). On universal oracle inequalities related to high-dimensional linear models. Ann. Statist. 38 2751-2780. · Zbl 1200.62074 · doi:10.1214/10-AOS803
[14] Gurdev, L. L., Dreischuh, T. N. and Stoyanov, D. V. (2002). High-range-resolution velocity-estimation techniques for coherent Doppler lidars with exponentially shaped laser pulses. Appl. Opt. 41 1741-1749.
[15] Harsdorf, S. and Reuter, R. (2000). Stable deconvolution of noisy lidar signals. In Proceedings of EARSeL-SIG-Workshop LIDAR . Dresden, FRG.
[16] Hoffmann, M. and Reiss, M. (2008). Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 310-336. · Zbl 1134.65038 · doi:10.1214/009053607000000721 · euclid:aos/1201877303
[17] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547-573. · Zbl 1046.62039 · doi:10.1111/j.1467-9868.2004.02056.x
[18] Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 58-109. · Zbl 1102.62318 · doi:10.1214/aos/1046294458
[19] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 137-170. · Zbl 1010.62029 · doi:10.1007/s004400100148
[20] Knapik, B. T., van der Vaart, A. W. and van Zanten, J. H. (2011). Bayesian inverse problems with Gaussian priors. Ann. Statist. 39 2626-2657. · Zbl 1232.62079 · doi:10.1214/11-AOS920
[21] Lepski, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454-466. · Zbl 0725.62075
[22] Lepski, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682-697. · Zbl 0776.62039 · doi:10.1137/1136085
[23] Lepski, O. V., Mammen, E. and Spokoiny, V. G. (1997). Optimal spatial adaptation to inhomogeneous smoothness: An approach based on kernel estimates with variable bandwidth selectors. Ann. Statist. 25 929-947. · Zbl 0885.62044 · doi:10.1214/aos/1069362731
[24] Lepski, O. V. and Spokoiny, V. G. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512-2546. · Zbl 0894.62041 · doi:10.1214/aos/1030741083
[25] Li, C. T. and Satta, R. (2011). On the location-dependent quality of the sensor pattern noise and its implication in multimedia forensics. In: 4 th International Conference on Imaging for Crime Detection and Prevention ( ICDP 2011). London, UK.
[26] Mair, B. A. and Ruymgaart, F. H. (1996). Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 1424-1444. · Zbl 0864.62020 · doi:10.1137/S0036139994264476
[27] Meyer, Y. (1992). Wavelets and Operators . Cambridge Univ. Press, Cambridge. · Zbl 0776.42019
[28] Miller, F. P., Vandome, A. F. and McBrewster, J. (2009). Amplitude Modulation . AlphaScript Publishing.
[29] Pensky, M. (2013). Supplement to “Spatially inhomogeneous linear inverse problems with possible singularities.” . · Zbl 1293.62014 · dx.doi.org
[30] Starck, J. L. and Pantin, E. (2002). Deconvolution in astronomy: A review. Publ. Astron. Soc. Pac. 114 1051-1069.
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