Bayesian inverse problems with unknown operators

Trabs, Mathias

doi:10.1088/1361-6420/aac3aa

Cited by 11 publications

(10 citation statements)

References 36 publications

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“…we haveα n ≤β andα n ≥β − C 2 log log n log n , (30) where C 2 = C 0 + C 1 . The notation C 2 will be used in all of the sections below.…”

Section: Analysis Of the Artificial Diagonal Problem Denote Mmentioning

confidence: 99%

“…where we employed lower bound estimate in (30). We should point out that s is assumed to be positive in Lemma 8 in [23].…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

“…Recently, by employing abstract tools from regularization theory, Agapiou and Mathé [2] study the posterior consistency by choosing a non-centered prior through empirical Bayes method. In 2018, Trabs [30] obtain optimal convergence rates up to logarithmic factors when the forward linear operator depends on an unknown parameter.…”

mentioning

confidence: 99%

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Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Jia

Peng

Gao

2021

IPI

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We investigate an empirical Bayesian nonparametric approach to a family of linear inverse problems with Gaussian prior and Gaussian noise. We consider a class of Gaussian prior probability measures with covariance operator indexed by a hyperparameter that quantifies regularity. By introducing two auxiliary problems, we construct an empirical Bayes method and prove that this method can automatically select the hyperparameter. In addition, we show that this adaptive Bayes procedure provides optimal contraction rates up to a slowly varying term and an arbitrarily small constant, without knowledge about the regularity index. Our method needs not the prior covariance, noise covariance and forward operator have a common basis in their singular value decomposition, enlarging the application range compared with the existing results. A simple simulation example is given that illustrates the effectiveness of the proposed method.

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“…we haveα n ≤β andα n ≥β − C 2 log log n log n , (30) where C 2 = C 0 + C 1 . The notation C 2 will be used in all of the sections below.…”

Section: Analysis Of the Artificial Diagonal Problem Denote Mmentioning

confidence: 99%

“…where we employed lower bound estimate in (30). We should point out that s is assumed to be positive in Lemma 8 in [23].…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

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Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Jia

Peng

Gao

2021

IPI

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“…Their estimator is given by a Lasso-type method modified so that the noise in the design matrix is taken care of. Finally, a different approach was introduced by Trabs [Tra18], who proposed a Bayesian method for the joint estimation of the unknown function f and the operator T . His setting is similar to that in [HR08] in that he observes the operator T up to additive random noise, instead of observing training data.…”

Section: Inverse Problems With Unknown Operatormentioning

confidence: 99%

Deep learning for inverse problems with unknown operator

Álamo¹

2021

Preprint

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We consider ill-posed inverse problems where the forward operator T is unknown, and instead we have access to training data consisting of functions f i and their noisy images T f i . This is a practically relevant and challenging problem which current methods are able to solve only under strong assumptions on the training set. Here we propose a new method that requires minimal assumptions on the data, and prove reconstruction rates that depend on the number of training points and the noise level. We show that, in the regime of "many" training data, the method is minimax optimal. The proposed method employs a type of convolutional neural networks (U-nets) and empirical risk minimization in order to "fit" the unknown operator. In a nutshell, our approach is based on two ideas: the first is to relate U-nets to multiscale decompositions such as wavelets, thereby linking them to the existing theory, and the second is to use the hierarchical structure of U-nets and the low number of parameters of convolutional neural nets to prove entropy bounds that are practically useful. A significant difference with the existing works on neural networks in nonparametric statistics is that we use them to approximate operators and not functions, which we argue is mathematically more natural and technically more convenient.

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“…Such models occur quite frequently in nonparametric statistics, see e.g. [9], [4], [19] and [35]. Other examples in this class of models concern deconvolution/ image deblurring from an unknown point-spread function (e.g.…”

Section: Introductionmentioning

confidence: 99%

Nonparametric estimation of the ability density in the Mixed-Effect Rasch Model

Kappus¹,

Liese²,

Meister³

2020

Electron. J. Statist.

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The Rasch model is widely used in the field of psychometrics when n persons under test answer m questions and the score, which describes the correctness of the answers, is given by a binary n × m-matrix. We consider the Mixed-Effect Rasch Model, in which the persons are chosen randomly from a huge population. The goal is to estimate the ability density of this population under nonparametric constraints, which turns out to be a statistical linear inverse problem with an unknown but estimable operator. Based on our previous result on asymptotic equivalence to a two-layer Gaussian model, we construct an estimation procedure and study its asymptotic optimality properties as n tends to infinity, as does m, but moderately with respect to n. Moreover numerical simulations are provided.

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Bayesian inverse problems with unknown operators

Cited by 11 publications

References 36 publications

Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Posterior contraction for empirical bayesian approach to inverse problems under non-diagonal assumption

Deep learning for inverse problems with unknown operator

Nonparametric estimation of the ability density in the Mixed-Effect Rasch Model

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