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Statistics and Computing

, Volume 29, Issue 6, pp 1185–1201 | Cite as

De-noising by thresholding operator adapted wavelets

  • Gene Ryan YooEmail author
  • Houman Owhadi
Article

Abstract

Donoho and Johnstone (Ann Stat 26(3):879–921, 1998) proposed a method from reconstructing an unknown smooth function u from noisy data \(u+\zeta \) by translating the empirical wavelet coefficients of \(u+\zeta \) towards zero. We consider the situation where the prior information on the unknown function u may not be the regularity of u but that of \( {\mathcal {L}}u\) where \({\mathcal {L}}\) is a linear operator (such as a PDE or a graph Laplacian). We show that the approximation of u obtained by thresholding the gamblet (operator adapted wavelet) coefficients of \(u+\zeta \) is near minimax optimal (up to a multiplicative constant), and with high probability, its energy norm (defined by the operator) is bounded by that of u up to a constant depending on the amplitude of the noise. Since gamblets can be computed in \({\mathcal {O}}(N {\text {polylog}} N)\) complexity and are localized both in space and eigenspace, the proposed method is of near-linear complexity and generalizable to nonhomogeneous noise.

Keywords

Probabilistic numerics Denoising Thresholding Wavelets Gamblet transform 

Notes

Acknowledgements

The authors gratefully acknowledges this work supported by the Air Force Office of Scientific Research and the DARPA EQUiPS Program under Award Number FA9550-16-1-0054 (Computational Information Games) and the Air Force Office of Scientific Research under Award Number FA9550-18-1-0271 (Games for Computation and Learning). We also thank two anonymous referees for detailed reviews and helpful comments.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.California Institute of TechnologyPasadenaUSA

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