Electronic Journal of Statistics

Multichannel boxcar deconvolution with growing number of channels

Marianna Pensky and Theofanis Sapatinas

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Abstract

We consider the problem of estimating the unknown response function in the multichannel deconvolution model with a boxcar-like kernel which is of particular interest in signal processing. It is known that, when the number of channels is finite, the precision of reconstruction of the response function increases as the number of channels M grow (even when the total number of observations n for all channels M remains constant) and this requires that the parameter of the channels form a Badly Approximable M-tuple.

Recent advances in data collection and recording techniques made it of urgent interest to study the case when the number of channels M=Mn grow with the total number of observations n. However, in real-life situations, the number of channels M=Mn usually refers to the number of physical devices and, consequently, may grow to infinity only at a slow rate as n. Unfortunately, existing theoretical results cannot be blindly applied to accommodate the case when M=Mn as n. This is due to the fact that, to the best of our knowledge, so far no one have studied the construction of a Badly Approximable M-tuple of a growing length on a specified interval, of a non-asymptotic length, of the real line, as M is growing. Therefore, this generalization requires non-trivial results in number theory.

When M=Mn grows slowly as n increases, we develop a procedure for the construction of a Badly Approximable M-tuple on a specified interval, of a non-asymptotic length, together with a lower bound associated with this M-tuple, which explicitly shows its dependence on M as M is growing. This result is further used for the evaluation of the L2-risk of the suggested adaptive wavelet thresholding estimator of the unknown response function and, furthermore, for the choice of the optimal number of channels M which minimizes the L2-risk.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 53-82.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1298644284

Digital Object Identifier
doi:10.1214/11-EJS597

Mathematical Reviews number (MathSciNet)
MR2773608

Zentralblatt MATH identifier
1274.62254

Subjects
Primary: 62G05: Estimation 11K60: Diophantine approximation [See also 11Jxx]
Secondary: 62G08: Nonparametric regression 35L05: Wave equation

Keywords
Adaptivity badly approximable tuples Besov spaces Diophantine approximation functional deconvolution Fourier analysis Meyer wavelets nonparametric estimation wavelet analysis

Citation

Pensky, Marianna; Sapatinas, Theofanis. Multichannel boxcar deconvolution with growing number of channels. Electron. J. Statist. 5 (2011), 53--82. doi:10.1214/11-EJS597. https://projecteuclid.org/euclid.ejs/1298644284


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References

  • [1] Abramovich, F. and Silverman, B.W. (1998). Wavelet decomposition approaches to statistical inverse problems., Biometrika 85 115–129.
  • [2] Casey, S.D. and Walnut, D.F. (1994). Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms., SIAM Review 36 537–577.
  • [3] Cassels, J.W.S. (1955). Simultaneous Diophantine approximation II., Proceedings of the London Mathematical Society 5 435–448.
  • [4] Cavalier, L. and Raimondo, M. (2007). Wavelet deconvolution with noisy eigenvalues., IEEE Transactions on Signal Processing 55 2414–2424.
  • [5] Chesneau, C. (2008). Wavelet estimation via block thresholding: a minimax study under, Lp-risk. Statistica Sinica 18 1007–1024.
  • [6] Davenport, H. (1962). A note on Diophantine approximation. In, Studies in Mathematical Analysis and Related Topics, (Eds. G. Szegö et al.), pp. 77–81, Stanford University Press, Stanford.
  • [7] De Canditiis, D. and Pensky, M. (2004). Discussion on the meeting on “Statistical Approaches to Inverse Problems”., Journal of the Royal Statistical Society, Series B 66 638–640.
  • [8] De Canditiis, D. and Pensky, M. (2006). Simultaneous wavelet deconvolution in periodic setting., Scandinavian Journal of Statistics 33 293–306.
  • [9] Donoho, D.L. (1995). Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition., Applied and Computational Harmonic Analysis 2 101–126.
  • [10] 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.
  • [11] Edixhoven, B. and Evertse, J.-H. (1993)., Diophantine Approximation and Abelian Varieties. Lecture Notes in Mathematics, Vol. 1566, Springer-Verlag, Berlin.
  • [12] Gradshtein, I.S. and Ryzhik, I.M. (1980)., Tables of Integrals, Series, and Products. Academic Press, New York.
  • [13] Harsdorf, S. and Reuter, R. (2000). Stable deconvolution of noisy lidar signals. In, Proceedings of EARSeL-SIG-Workshop LIDAR, Dresden/FRG, June 16–17.
  • [14] Johnstone, I.M. (2002)., Function Estimation in Gaussian Noise: Sequence Models. Unpublished Monograph. (http://www-stat.stanford.edu/~imj/)
  • [15] 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).
  • [16] Johnstone, I.M. and Raimondo, M. (2004). Periodic boxcar deconvolution and Diophantine approximation., Annals of Statistics 32 1781–1804.
  • [17] Kalifa, J. and Mallat, S. (2003). Thresholding estimators for linear inverse problems and deconvolutions., Annals of Statistics 31 58–109.
  • [18] Kerkyacharian, G., Picard, D. and Raimondo, M. (2007). Adaptive boxcar deconvolution on full Lebesgue measure sets., Statistica Sinica 7 317–340.
  • [19] Lang, S. (1966), Introduction to Diophantine Approximations. Springer-Verlag, New York.
  • [20] Mallat, S.G. (1999)., A Wavelet Tour of Signal Processing, 2nd Edition, Academic Press, San Diego.
  • [21] Masser, D., Nesterenko, Yu.V., Schlickewei, H.P., Schmidt, W.M. and Waldschmidt, M. (2003)., Diophantine Approximation. Lecture Notes in Mathematics, Vol. 1819, Springer-Verlag, Berlin.
  • [22] Meyer, Y. (1992)., Wavelets and Operators. Cambridge University Press, Cambridge.
  • [23] 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.
  • [24] Park, Y.J., Dho, S.W. and Kong, H.J. (1997). Deconvolution of long-pulse lidar signals with matrix formulation., Applied Optics 36 5158–5161.
  • [25] Pensky, M. and Sapatinas, T. (2009). Functional deconvolution in a periodic case: uniform case., Annals of Statistics 37 73–104.
  • [26] Pensky, M. and Sapatinas, T. (2010). On convergence rates equivalency and sampling strategies in functional deconvolution models., Annals of Statistics 38 1793–1844.
  • [27] Schmidt, W. (1969). Badly approximable systems of linear forms., Journal of Number Theory 1 139–154.
  • [28] Schmidt, W. (1980)., Diophantine Approximation. Lecture Notes in Mathematics, Vol. 785, Springer-Verlag, Berlin.
  • [29] Schmidt, W. (1991)., Diophantine Approximations and Diophantine Equations. Lecture Notes in Mathematics, Vol. 1467, Springer-Verlag, Berlin.