## Electronic Journal of Statistics

### Solution of linear ill-posed problems by model selection and aggregation

#### Abstract

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.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 1822-1841.

Dates
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.ejs/1528769121

Digital Object Identifier
doi:10.1214/18-EJS1447

Mathematical Reviews number (MathSciNet)
MR3813598

Zentralblatt MATH identifier
06886386

Subjects
Primary: 62G05: Estimation
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

#### Citation

Abramovich, Felix; De Canditiis, Daniela; Pensky, Marianna. Solution of linear ill-posed problems by model selection and aggregation. Electron. J. Statist. 12 (2018), no. 1, 1822--1841. doi:10.1214/18-EJS1447. https://projecteuclid.org/euclid.ejs/1528769121

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