Electronic Journal of Statistics

Smooth hyperbolic wavelet deconvolution with anisotropic structure

J.R. Wishart

Full-text: Open access


This paper considers a deconvolution regression problem in a multivariate setting with anisotropic structure and constructs an estimator of the function of interest using the hyperbolic wavelet basis. The deconvolution structure assumed is an anisotropic version of the smooth type (either regular-smooth or super-smooth). The function of interest is assumed to belong to a Besov space with anisotropic smoothness. Global performances of the presented hyperbolic wavelet estimators is measured by obtaining upper bounds on convergence rates in the $\mathscr{L}^{p}$-risk with $1\le p\le 2$ and $1\le p<\infty $ in the regular-smooth and super-smooth cases respectively. The results are compared and contrasted with existing convergence results in the literature.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 1694-1716.

Received: May 2018
First available in Project Euclid: 24 April 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G05: Estimation 62G20: Asymptotic properties
Secondary: 60J65: Brownian motion [See also 58J65] 65T60: Wavelets

Besov spaces Brownian sheet deconvolution Fourier analysis Meyer wavelets hyperbolic wavelet analysis anisotropic

Creative Commons Attribution 4.0 International License.


Wishart, J.R. Smooth hyperbolic wavelet deconvolution with anisotropic structure. Electron. J. Statist. 13 (2019), no. 1, 1694--1716. doi:10.1214/19-EJS1557. https://projecteuclid.org/euclid.ejs/1556071301

Export citation


  • [1] Abramovich, F. and Silverman, B. W. (1998). Wavelet decomposition approaches to statistical inverse problems., Biometrika 85 115–129.
  • [2] Abry, P., Roux, S. G., Wendt, H., Messier, P., Klein, A. G., Tremblay, N., Borgnat, P., Jaffard, S., Vedel, B., Coddington, J. and Daffner, L. A. (2015). Multiscale Anisotropic Texture Analysis and Classification of Photographic Prints: Art scholarship meets image processing algorithms., IEEE Signal Processing Magazine 32 18–27.
  • [3] Autin, F., Claeskens, G. and Freyermuth, J. M. (2014). Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach., Applied and Computational Harmonic Analysis 36 239–255.
  • [4] Autin, F., Claeskens, G. and Freyermuth, J.-m. (2015). Asymptotic performance of projection estimators in standard and hyperbolic wavelet bases., Electronic Journal of Statistics 9 1852–1883.
  • [5] Bekmaganbetov, K. A. and Nursultanov, E. D. (2009). Embedding theorems for anisotropic Besov spaces., Izvestiya: Mathematics 73 655–668.
  • [6] Benhaddou, R. (2017). On minimax convergence rates under Lp-risk for the anisotropic functional deconvolution model., Statistics and Probability Letters 130 120–125.
  • [7] Benhaddou, R., Pensky, M. and Picard, D. (2013). Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates., Electronic Journal of Statistics 7 1686–1715.
  • [8] Benhaddou, R., Pensky, M. and Rajapakshage, R. (2019). Anisotropic functional Laplace deconvolution., Journal of Statistical Planning and Inference 199 271–285.
  • [9] Benhaddou, R., Kulik, R., Pensky, M. and Sapatinas, T. (2014). Multichannel deconvolution with long-range dependence: A minimax study., Journal of Statistical Planning and Inference 148 1–19.
  • [10] Besov, O. V., Il’in, V. P. and Nikol’skiĭ, S. M. (1978)., Integral Representation of Functions and Imbedding Theorems, Vol ii ed. V. H. Winston & Sons, Washington, D.C.; Halsted Press John Wiley & Sons, New York-Toronto, Ont.-London.
  • [11] Daubechies, I. (1992)., Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics.
  • [12] Deliège, A., Kleyntssens, T. and Nicolay, S. (2017). Mars topography investigated through the wavelet leaders method: A multidimensional study of its fractal structure., Planetary and Space Science 136 46–58.
  • [13] DeVore, R. A., Konyagin, S. V. and Temlyakov, V. N. (1998). Hyperbolic Wavelet Approximation., Constructive Approximation 14 1–26.
  • [14] Donoho, D. L. (1993). Unconditional Bases are Optimal Bases for Data Compression and for Statistical Estimation., Applied and Computational Harmonic Analysis 1 100–115.
  • [15] Donoho, D. (1995). Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition., Applied and Computational Harmonic Analysis 2 101–126.
  • [16] Donoho, D. L. and Raimondo, M. E. (2004). A fast wavelet algorithm for image deblurring. In, Proc. of 12th Computational Techniques and Applications Conference CTAC-2004 46 29–46.
  • [17] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet Shrinkage: Asymptopia?, Journal of the Royal Statistical Society. Series B (Methodological) 57 301–369.
  • [18] Fan, J. (1991). On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems., The Annals of Statistics 19 1257–1272.
  • [19] Fan, J. and Koo, J.-y. (2002). Wavelet deconvolution., IEEE Transactions on Information Theory 48 734–747.
  • [20] Farouj, Y., Freyermuth, J.-m., Navarro, L., Clausel, M. and Delachartre, P. (2017). Hyperbolic Wavelet-Fisz Denoising for a Model Arising in Ultrasound Imaging., IEEE Transactions on Computational Imaging 3 1–10.
  • [21] Heping, W. (2004). Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets., Journal of Mathematical Analysis and Applications 291 698–715.
  • [22] Johnstone, I. M. (1999). Wavelet shrinkage for correlated data and inverse problems: Adaptivity results., Statistica Sinica 9 51–83.
  • [23] 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 (Statistical Methodology) 66 547–573.
  • [24] Kerkyacharian, G., Lepski, O. and Picard, D. (2001). Nonlinear estimation in anisotropic multi-index denoising., Probability Theory and Related Fields 121 137–170.
  • [25] Kerkyacharian, G., Lepski, O. and Picard, D. (2008). Nonlinear Estimation in Anisotropic Multi-Index Denoising. Sparse Case., Theory of Probability & Its Applications 52 58–77.
  • [26] Kerkyacharian, G. and Picard, D. (2000). Thresholding algorithms, maxisets and well-concentrated bases., Test 9 283–344.
  • [27] Kerkyacharian, G. and Picard, D. (2003). Entropy, Universal Coding, Approximation, and Bases Properties., Constructive Approximation 20 1–37.
  • [28] Kulik, R. and Raimondo, M. $L^p$-Wavelet regression with correlated errors and inverse problems., Statistica Sinica 4 1479–1489.
  • [29] Kulik, R., Sapatinas, T. and Wishart, J. R. (2015). Multichannel deconvolution with long range dependence: Upper bounds on the $L^p$-risk $(1\le p<\infty )$., Applied and Computational Harmonic Analysis 38 357–384.
  • [30] Meyer, Y. and Salinger, D. H. (1993)., Wavelets and Operators 1, 1st ed. Cambridge University Press.
  • [31] Neumann, M. H. (2000). Multivariate wavelet thresholding in anisotropic function spaces., Statistica Sinica 10 399–432.
  • [32] Neumann, M. H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra., The Annals of Statistics 25 38–76.
  • [33] Pensky, M. and Sapatinas, T. (2009). Functional Deconvolution in a Periodic Setting: Uniform Case., The Annals of Statistics 37 73–104.
  • [34] Pensky, M. and Sapatinas, T. (2010). On convergence rates equivalency and sampling strategies in functional deconvolution models., The Annals of Statistics 38 1793–1844.
  • [35] Pensky, M. and Sapatinas, T. (2011). Multichannel boxcar deconvolution with growing number of channels., Electronic Journal of Statistics 5 53–82.
  • [36] Petsa, A. and Sapatinas, T. (2009). Minimax convergence rates under the -risk in the functional deconvolution model., Statistics & Probability Letters 79 1568–1576.
  • [37] Proksch, K., Bissantz, N. and Dette, H. (2012). A note on asymptotic uniform confidence bands in a multivariate statistical deconvolution problem. In, ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics (T. E. Simos, G. Psihoyios, C. Tsitouras and Z. Anastassi, eds.) 1479 438–441. AIP Publishing.
  • [38] Proksch, K., Bissantz, N. and Dette, H. (2015). Confidence bands for multivariate and time dependent inverse regression models., Bernoulli 21 144–175.
  • [39] Remenyi, N., Nicolis, O., Nason, G. and Vidakovic, B. (2014). Image Denoising With 2D Scale-Mixing Complex Wavelet Transforms., IEEE Transactions on Image Processing 23 5165–5174.
  • [40] Richard, F. J. P. (2018). Anisotropy of Hölder Gaussian random fields: characterization, estimation, and application to image textures., Statistics and Computing 28 1155–1168.
  • [41] Roux, S. G., Clausel, M., Vedel, B., Jaffard, S. and Abry, P. (2013). Self-similar anisotropic texture analysis: The hyperbolic wavelet transform contribution., IEEE Transactions on Image Processing 22 4353–4363.
  • [42] Triebel, H. (2006)., Theory of Function Spaces III. Monographs in Mathematics 100. Birkhäuser Basel.
  • [43] Wang, Y. (1997). Minimax estimation via wavelets for indirect long-memory data., Journal of Statistical Planning and Inference 64 45–55.
  • [44] Wishart, J. R. (2013). Wavelet deconvolution in a periodic setting with long-range dependent errors., Journal of Statistical Planning and Inference 143 867–881.