## Electronic Journal of Statistics

### Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates

#### Abstract

In the present paper we consider the problem of estimating a periodic $(r+1)$-dimensional function $f$ based on observations from its noisy convolution. We construct a wavelet estimator of $f$, derive minimax lower bounds for the $L^{2}$-risk when $f$ belongs to a Besov ball of mixed smoothness and demonstrate that the wavelet estimator is adaptive and asymptotically near-optimal within a logarithmic factor, in a wide range of Besov balls. We prove in particular that choosing this type of mixed smoothness leads to rates of convergence which are free of the “curse of dimensionality” and, hence, are higher than usual convergence rates when $r$ is large.

The problem studied in the paper is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Indeed, we show that unless the function $f$ is very smooth in the direction of the profiles, very spatially inhomogeneous along the other direction and the number of profiles is very limited, the functional deconvolution solution has a much better precision compared to a combination of $M$ solutions of separate convolution equations. A limited simulation study in the case of $r=1$ confirms theoretical claims of the paper.

#### Article information

Source
Electron. J. Statist., Volume 7 (2013), 1686-1715.

Dates
First available in Project Euclid: 26 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1372251196

Digital Object Identifier
doi:10.1214/13-EJS820

Mathematical Reviews number (MathSciNet)
MR3080407

Zentralblatt MATH identifier
1294.62057

Subjects
Primary: 62G05: Estimation
Secondary: 62G08: Nonparametric regression 62P35: Applications to physics

#### Citation

Benhaddou, Rida; Pensky, Marianna; Picard, Dominique. Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates. Electron. J. Statist. 7 (2013), 1686--1715. doi:10.1214/13-EJS820. https://projecteuclid.org/euclid.ejs/1372251196

#### References

• [1] Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems., Biometrika. 85 115-129.
• [2] Berkhout, A. J. (1986). The seismic method in the search for oil and gas: Current techniques and future developments., Proc. IEEE. 74 1133-1159.
• [3] Bunea, F.,Tsybakov, A. and Wegkamp, M. H. (2007). Aggregation for Gaussian regression., Ann. Statist. 35 1674-1697.
• [4] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage., Biometrika. 81 425-456.
• [5] Donoho, D. L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition., Applied and Computational Harmonic Analysis. 2 101-126.
• [6] Donoho, D. L. and Raimondo, M. (2004). Translation invariant deconvolution in a periodic setting., International Journal of Wavelets, Multiresolution and Information Processing. 14 415-432.
• [7] Fan, J. and Koo, J. (2002). Wavelet deconvolution., IEEE Transactions on Information Theory. 48 734-747.
• [8] Heimer, A. and Cohen, I. (2008). Multichannel blind seismic deconvolution using dynamic programming., Signal Processing. 88 1839-1851
• [9] Heping, W. (2004). Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets, J. Math. Anal. Appl. 291 698-715.
• [10] Holdai, V. and Korostelev, A. (2008). Image Reconstruction in Multi-Channel Model under Gaussian Noise., Math. Meth. Statist. 17 198-208.
• [11] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting., Journal of the Royal Statistical Society. Series B, 66 547-573 (with discussion, 627-657).
• [12] Johnstone, I. M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation., Annals of Statistics. 32 1781-1804.
• [13] Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions., Annals of Statistics. 31 58-109.
• [14] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising., Probab. Theory Relat. Fields. 121 137-170.
• [15] Kerkyacharian, G.,Lepski, O. and Picard, D. (2008). Nonlinear Estimation in Anisotropic Multi-Index Denoising. Sparse Case., Theory Probab. Appl. 52 58-77.
• [16] Kerkyacharian, G., Picard, D. and Raimondo, M. (2007). Adaptive boxcar deconvolution on full Lebesgue measure sets., Statistica Sinica. 7 317-340.
• [17] Neelamani, R., Choi, H. and Baraniuk, R. (2004). Forward: Fourier-wavelet regularized deconvolution for ill-conditioned systems., IEEE Transactions on Signal Processing. 52 418-433.
• [18] Nikolskii S.M. (1975)., Approximation of functions of several variables and imbedding theorems (Russian). Sec. ed., Moskva, Nauka 1977 English translation of the first ed., Berlin.
• [19] Pensky, M. and Sapatinas, T. (2009). Functional Deconvolution in a Periodic Setting: Uniform Case., Annals of Statistics. 37 73-104.
• [20] Pensky, M. and Sapatinas, T. (2010). On Convergence Rates Equivalency and Sampling Strategies in a Functional Deconvolution Model., Annals of Statistics. 38 1793-1844.
• [21] Pensky, M. and Sapatinas, T. (2011). Multichannel Boxcar Deconvolution with Growing Number of Channels., Electronic Journal of Statistics. 5 53-82.
• [22] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution., Annals of Statistics. 27 2033-2053.
• [23] Robinson, E. A. (1999)., Seismic Inversion and Deconvolution: Part B: Dual-Sensor Technology. Elsevier, Oxford.
• [24] Robinson, E. A., Durrani, T. S. and Peardon, L. G. (1986)., Geophysical Signal Processing. Prentice-Hall, London.
• [25] Tsybakov, A. B. (2008)., Introduction to Nonparametric Estimation. Springer, New York.
• [26] Walter, G. and Shen, X. (1999). Deconvolution using Meyer wavelets., Journal of Integral Equations and Applications. 11 515-534.
• [27] Wason, C. B., Black, J. L. and King, G. A. (1984). Seismic modeling and inversion., Proc. IEEE. 72 1385-1393.