The Annals of Statistics

Spatially inhomogeneous linear inverse problems with possible singularities

Marianna Pensky

Full-text: Open access

Abstract

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 Lepski [Theory Probab. Appl. 35 (1990) 454–466 and 36 (1991) 682–697] 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.

Article information

Source
Ann. Statist., Volume 41, Number 5 (2013), 2668-2697.

Dates
First available in Project Euclid: 19 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1384871349

Digital Object Identifier
doi:10.1214/13-AOS1166

Mathematical Reviews number (MathSciNet)
MR3161441

Zentralblatt MATH identifier
1293.62014

Subjects
Primary: 62C10: Bayesian problems; characterization of Bayes procedures 65J10: Equations with linear operators (do not use 65Fxx)
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Keywords
Linear inverse problems inhomogeneous minimax convergence rates singularity

Citation

Pensky, Marianna. Spatially inhomogeneous linear inverse problems with possible singularities. Ann. Statist. 41 (2013), no. 5, 2668--2697. doi:10.1214/13-AOS1166. https://projecteuclid.org/euclid.aos/1384871349


Export citation

References

  • Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 115–129.
  • 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. To appear.
  • 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.
  • Cavalier, L. and Golubev, Y. (2006). Risk hull method and regularization by projections of ill-posed inverse problems. Ann. Statist. 34 1653–1677.
  • Cavalier, L., Golubev, G. K., Picard, D. and Tsybakov, A. B. (2002). Oracle inequalities for inverse problems. Ann. Statist. 30 843–874.
  • Cohen, A., Hoffmann, M. and Reiß, M. (2004). Adaptive wavelet Galerkin methods for linear inverse problems. SIAM J. Numer. Anal. 42 1479–1501 (electronic).
  • Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet–vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 101–126.
  • Engl, H. W., Hanke, M. and Neubauer, A. (1996). Regularization of Inverse Problems. Kluwer Academic, Dordrecht.
  • Gaïffas, S. (2005). Convergence rates for pointwise curve estimation with a degenerate design. Math. Methods Statist. 14 1–27.
  • Gaïffas, S. (2007a). Sharp estimation in sup norm with random design. Statist. Probab. Lett. 77 782–794.
  • Gaïffas, S. (2007b). On pointwise adaptive curve estimation based on inhomogeneous data. ESAIM Probab. Stat. 11 344–364 (electronic).
  • Gaïffas, S. (2009). Uniform estimation of a signal based on inhomogeneous data. Statist. Sinica 19 427–447.
  • Golubev, Y. (2010). On universal oracle inequalities related to high-dimensional linear models. Ann. Statist. 38 2751–2780.
  • 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.
  • Harsdorf, S. and Reuter, R. (2000). Stable deconvolution of noisy lidar signals. In Proceedings of EARSeL-SIG-Workshop LIDAR. Dresden, FRG.
  • Hoffmann, M. and Reiss, M. (2008). Nonlinear estimation for linear inverse problems with error in the operator. Ann. Statist. 36 310–336.
  • 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.
  • Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 58–109.
  • Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising. Probab. Theory Related Fields 121 137–170.
  • 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.
  • Lepski, O. V. (1990). A problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl. 35 454–466.
  • Lepski, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Theory Probab. Appl. 36 682–697.
  • 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.
  • Lepski, O. V. and Spokoiny, V. G. (1997). Optimal pointwise adaptive methods in nonparametric estimation. Ann. Statist. 25 2512–2546.
  • Li, C. T. and Satta, R. (2011). On the location-dependent quality of the sensor pattern noise and its implication in multimedia forensics. In: 4th International Conference on Imaging for Crime Detection and Prevention (ICDP 2011). London, UK.
  • Mair, B. A. and Ruymgaart, F. H. (1996). Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 1424–1444.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge Univ. Press, Cambridge.
  • Miller, F. P., Vandome, A. F. and McBrewster, J. (2009). Amplitude Modulation. AlphaScript Publishing.
  • Pensky, M. (2013). Supplement to “Spatially inhomogeneous linear inverse problems with possible singularities.” DOI:10.1214/13-AOS1166SUPP.
  • Starck, J. L. and Pantin, E. (2002). Deconvolution in astronomy: A review. Publ. Astron. Soc. Pac. 114 1051–1069.

Supplemental materials