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De-noising by thresholding operator adapted wavelets

Yoo, Gene Ryan and Owhadi, Houman (2019) De-noising by thresholding operator adapted wavelets. Statistics and Computing, 29 (6). pp. 1185-1201. ISSN 0960-3174. doi:10.1007/s11222-019-09893-x.

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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.

Item Type:Article
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URLURL TypeDescription ReadCube access Paper
Yoo, Gene Ryan0000-0002-5319-5599
Owhadi, Houman0000-0002-5677-1600
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.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-16-1-0054
Air Force Office of Scientific Research (AFOSR)FA9550-18-1-0271
Subject Keywords:Probabilistic numerics; Denoising; Thresholding; Wavelets; Gamblet transform
Issue or Number:6
Record Number:CaltechAUTHORS:20190923-104545857
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Official Citation:Yoo, G.R. & Owhadi, H. Stat Comput (2019) 29: 1185.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98797
Deposited By: Tony Diaz
Deposited On:23 Sep 2019 17:51
Last Modified:16 Nov 2021 17:41

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