Optimal bounds for inverse problems with Jacobi-type eigenfunctions - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Statistica Sinica Année : 2009

Optimal bounds for inverse problems with Jacobi-type eigenfunctions

Résumé

We consider inverse problems where one wishes to recover an unknown function from the observation of a transformation of it by a linear operator, corrupted by an additive white noise perturbation. We assume that the operator admits a singular value decomposition where the eigenvalues decay in a polynomial way, and where Jacobi polynomials appear as eigenfunctions. This includes, as an application, the well known Wicksell's problem. We determine the asymptotic rate of the minimax risk for this model in a wide framework, considering (Lp) losses (1 < p < \infty), and Besov-like regularity spaces. We draw a comparison with the minimax rates of the deconvolution problem, which appears as a critical case of the Jacobi-type rates. We also establish some new results on the needlets introduced by Petrushev and Xu (2005) which appear as essential tools in this setting.
Fichier principal
Vignette du fichier
newrates.pdf (343.27 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00133830 , version 1 (27-02-2007)
hal-00133830 , version 2 (21-10-2013)

Identifiants

  • HAL Id : hal-00133830 , version 2

Citer

Thomas Willer. Optimal bounds for inverse problems with Jacobi-type eigenfunctions. Statistica Sinica, 2009, 19 (2), pp.785-800. ⟨hal-00133830v2⟩
125 Consultations
93 Téléchargements

Partager

Gmail Facebook X LinkedIn More