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2018 Solution of linear ill-posed problems by model selection and aggregation
Felix Abramovich, Daniela De Canditiis, Marianna Pensky
Electron. J. Statist. 12(1): 1822-1841 (2018). DOI: 10.1214/18-EJS1447


We consider a general statistical linear inverse problem, where the solution is represented via a known (possibly overcomplete) dictionary that allows its sparse representation. We propose two different approaches. A model selection estimator selects a single model by minimizing the penalized empirical risk over all possible models. By contrast with direct problems, the penalty depends on the model itself rather than on its size only as for complexity penalties. A Q-aggregate estimator averages over the entire collection of estimators with properly chosen weights. Under mild conditions on the dictionary, we establish oracle inequalities both with high probability and in expectation for the two estimators. Moreover, for the latter estimator these inequalities are sharp. The proposed procedures are implemented numerically and their performance is assessed by a simulation study.


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Felix Abramovich. Daniela De Canditiis. Marianna Pensky. "Solution of linear ill-posed problems by model selection and aggregation." Electron. J. Statist. 12 (1) 1822 - 1841, 2018.


Received: 1 October 2017; Published: 2018
First available in Project Euclid: 12 June 2018

zbMATH: 06886386
MathSciNet: MR3813598
Digital Object Identifier: 10.1214/18-EJS1447

Primary: 62G05
Secondary: 62C10

Keywords: Aggregation , ill-posed linear inverse problem , Model selection , Oracle inequality , overcomplete dictionary


Vol.12 • No. 1 • 2018
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