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Weyl eigenvalue asymptotics and sharp adaptation on vector bundles

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  • Kim, Peter T.
  • Koo, Ja-Yong
  • Luo, Zhi-Ming
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    Abstract

    This paper examines the estimation of an indirect signal embedded in white noise on vector bundles. It is found that the sharp asymptotic minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus when the linear operator has polynomial decay, recovery of the signal is polynomial where the exact minimax constant and rate are determined. Adaptive sharp estimation is carried out using a blockwise shrinkage estimator. Application to the spherical deconvolution problem for the polynomially bounded case is made.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 100 (2009)
    Issue (Month): 9 (October)
    Pages: 1962-1978

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    Handle: RePEc:eee:jmvana:v:100:y:2009:i:9:p:1962-1978

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    Related research

    Keywords: Eigenstructure Laplacian Pinsker-Weyl bound Riemannian geometry Sobolev ellipsoid Spectral geometry Weyl constant;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Bissantz, Nicolai & Hohage, T. & Munk, Axel & Ruymgaart, F., 2007. "Convergence rates of general regularization methods for statistical inverse problems and applications," Technical Reports 2007,04, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    2. Goldenshluger, Alexander, 2002. "Density Deconvolution in the Circular Structural Model," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 360-375, May.
    3. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    4. Koo, Ja-Yong & Kim, Peter T., 2008. "Sharp adaptation for spherical inverse problems with applications to medical imaging," Journal of Multivariate Analysis, Elsevier, vol. 99(2), pages 165-190, February.
    5. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    6. Kim, Peter T. & Koo, Ja-Yong & Park, Heon Jin, 2004. "Sharp minimaxity and spherical deconvolution for super-smooth error distributions," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 384-392, August.
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    Cited by:
    1. Durastanti, Claudio & Geller, Daryl & Marinucci, Domenico, 2012. "Adaptive nonparametric regression on spin fiber bundles," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 16-38, February.
    2. Hielscher, Ralf, 2013. "Kernel density estimation on the rotation group and its application to crystallographic texture analysis," Journal of Multivariate Analysis, Elsevier, vol. 119(C), pages 119-143.

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