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

Felix Abramovich, Daniela De Canditiis, Marianna Pensky

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)1822-1841
Number of pages20
JournalElectronic Journal of Statistics
Issue number1
StatePublished - 2018


FundersFunder number
National Science FoundationDMS-1407475, DMS-1712977
Directorate for Mathematical and Physical Sciences1407475, 1712977


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


    Dive into the research topics of 'Solution of linear ill-posed problems by model selection and aggregation'. Together they form a unique fingerprint.

    Cite this