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Denoising strategies for general finite frames

Author

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  • De Canditiis, D.
  • Pensky, M.
  • Wolfe, P.J.

Abstract

Overcomplete representations such as wavelets and windowed Fourier expansions have become mainstays of modern statistical data analysis. In the present work, in the context of general finite frames, we derive an oracle expression for the mean quadratic risk of a linear diagonal de-noising procedure which immediately yields the optimal linear diagonal estimator. Moreover, we obtain an expression for an unbiased estimator of the risk of any smooth shrinkage rule. This last result motivates a set of practical estimation procedures for general finite frames that can be viewed as the generalization of the classical procedures for orthonormal bases. A simulation study verifies the effectiveness of the proposed procedures with respect to the classical ones and confirms that the correlations induced by frame structure should be explicitly treated to yield an improvement in estimation precision.

Suggested Citation

  • De Canditiis, D. & Pensky, M. & Wolfe, P.J., 2018. "Denoising strategies for general finite frames," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 147(C), pages 90-99.
    • Handle: RePEc:eee:matcom:v:147:y:2018:i:c:p:90-99
    DOI: 10.1016/j.matcom.2017.02.005
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    References listed on IDEAS

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    1. Patrick J. Wolfe & Simon J. Godsill & Wee‐Jing Ng, 2004. "Bayesian variable selection and regularization for time–frequency surface estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 575-589, August.
    2. De Canditiis, Daniela, 2014. "A frame based shrinkage procedure for fast oscillating functions," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 142-150.
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