Solution Of Linear Ill-Posed Problems By Model Selection And Aggregation

Keywords

Aggregation; Ill-posed linear inverse problem; Model selection; Oracle inequality; Overcomplete dictionary

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.

Publication Date

1-1-2018

Publication Title

Electronic Journal of Statistics

Volume

12

Issue

1

Number of Pages

1822-1841

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1214/18-EJS1447

Socpus ID

85048512611 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/85048512611

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