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


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    1. 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.
    2. 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.
    3. Jussi Klemela, 1999. "Asymptotic Minimax Risk for the White Noise Model on the Sphere," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 26(3), pages 465-473.
    4. Goldenshluger, Alexander, 2002. "Density Deconvolution in the Circular Structural Model," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 360-375, May.
    5. Healy, Dennis M. & Hendriks, Harrie & Kim, Peter T., 1998. "Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 1-22, October.
    6. 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.
    7. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
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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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