The dictionary approach for spherical deconvolution

Rivoirard, Vincent; Pham Ngoc, Thanh Mai

Rivoirard, Vincent; Pham Ngoc, Thanh Mai (2013), The dictionary approach for spherical deconvolution, Journal of Multivariate Analysis, 115, p. 138-156. http://dx.doi.org/10.1016/j.jmva.2012.08.011

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00616519/fr/

Date

2013

Journal name

Journal of Multivariate Analysis

Volume

115

Publisher

Elsevier

Pages

138-156

Abstract (EN)

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 approches 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 very satisfying results in the numerical study when compared with other procedures.

Subjects / Keywords

Second generation wavelets; Sparsity; Calibration; Oracle inequalities; Lasso estimate; Dictionary; Density deconvolution