Oracle inequalities for inverse problems

Cavalier, Laurent; Golubev, G. K.; Picard, Dominique; Tsybakov, Alexandre B.

doi:10.1214/aos/1028674843

Cited by 133 publications

(152 citation statements)

References 27 publications

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“…We refer the reader to the paper of Kneip (1993) for an extensive bibliography on data-driven choice of smoothing parameters. More recent references are (Donoho and Johnstone 1995;Birgé and Massart 2001;Cavalier, Golubev, Picard, and Tsybakov 2002).…”

Section: It Follows Thatmentioning

confidence: 99%

Adaptive minimax estimation of a fractional derivative

Enikeeva¹

2006

Statistics & Probability Letters

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Section: It Follows Thatmentioning

confidence: 99%

Adaptive minimax estimation of a fractional derivative

Enikeeva¹

2006

Statistics & Probability Letters

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“…This estimator is adaptive both over Sobolev and analytic scales. In [10] the data-driven choice of the regularizing parameters is based on unbiased risk estimation. The authors consider projection estimators and derive the corresponding oracle inequalities.…”

Section: Introductionmentioning

confidence: 99%

Bayes procedures for adaptive inference in inverse problems for the white noise model

Knapik¹,

Szabó

Vaart

et al. 2015

Probab. Theory Relat. Fields

142

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We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies regularity. We prove that both methods we consider succeed in automatically selecting this parameter optimally, resulting in optimal convergence rates for truths with Sobolev or analytic "smoothness", without using knowledge about this regularity. Both methods are illustrated by simulation examples.

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“…Projection methods, which are defined as solutions of (1) restricted on finite dimensional subspaces H N and K N (of dimension N ) also give rise to attractive approximations of f , by properly choosing the subspaces and the tuning parameter N (Dicken and Maass [10], Mathe and Pereversseh [31] together with their non linear counterparts Cavalier and Tsybakov [7], Cavalier et al [6], Tsybakov [40], Goldenschluger and Pereversev [19], Efromovich and Kolchinskii [16]. In the case where H = K, and K is a self adjoint operator, the system is particularly simple to solve since the restricted operator K N is symmetric positive definite.…”

Section: Projection Methodsmentioning

confidence: 99%

Estimation in inverse problems and second-generation wavelets

Kerkyacharian¹,

Picard²

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We consider the problem of recovering a function f , when we receive a blurred (by a linear operator) and noisy version : Yε = Kf + εẆ . We will have as guides 2 famous examples of such inverse problems : the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It can't be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising we have to 'inverse the operator' K, which operation not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem, there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable : this is the Singular Value Decomposition basis. On the other hand, the SVD basis might be difficult to determine and manipulate numerically, it also might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations, the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures trying to keep the advantages of localisation without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD) using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithm factors for any Lp loss function, and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD bases lies in the range of Jacobi polynomials), we prove that quite a similar algorithm (NEED-VD) can be performed provided replacing the standard wavelet system by a second generation wavelet-type basis : the Needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I). It has its own interest and will act as a guide for understanding the 'true' inverse models for a reader which is unfamiliar with nonpar...

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Oracle inequalities for inverse problems

Cited by 133 publications

References 27 publications

Adaptive minimax estimation of a fractional derivative

Adaptive minimax estimation of a fractional derivative

Bayes procedures for adaptive inference in inverse problems for the white noise model

Estimation in inverse problems and second-generation wavelets

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