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Published November 2019 | Submitted
Journal Article Open

De-noising by thresholding operator adapted wavelets


Donoho and Johnstone (Ann Stat 26(3):879–921, 1998) proposed a method from reconstructing an unknown smooth function u from noisy data u+ζ by translating the empirical wavelet coefficients of u+ζ 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 Lu where 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+ζ 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 O(NpolylogN) complexity and are localized both in space and eigenspace, the proposed method is of near-linear complexity and generalizable to nonhomogeneous noise.

Additional Information

© 2019 Springer Science+Business Media, LLC, part of Springer Nature. First Online: 21 September 2019. 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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