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The dictionary approach for spherical deconvolution

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  • Pham Ngoc, Thanh Mai
  • Rivoirard, Vincent

Abstract

We consider the problem of estimating a density of probability from indirect data in the spherical convolution model. We aim at building an estimate of the unknown density as a linear combination of functions of an overcomplete dictionary. The procedure is devised through a well-calibrated ℓ1-penalized criterion. The spherical deconvolution setting has been barely studied so far, and the two main approaches to this problem, namely the SVD and the hard thresholding ones considered only one basis at a time. The dictionary approach allows to combine various bases and thus enhances estimates sparsity. We provide an oracle inequality under global coherence assumptions. Moreover, the calibrated procedure that we put forward gives quite satisfying results in the numerical study when compared with other procedures.

Suggested Citation

  • Pham Ngoc, Thanh Mai & Rivoirard, Vincent, 2013. "The dictionary approach for spherical deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 138-156.
    • Handle: RePEc:eee:jmvana:v:115:y:2013:i:c:p:138-156
    DOI: 10.1016/j.jmva.2012.08.011
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    References listed on IDEAS

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    1. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    2. Kim, Peter T. & Koo, Ja-Yong, 2002. "Optimal Spherical Deconvolution," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 21-42, January.
    3. 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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