Convex regularization in statistical inverse learning problems

Tatiana Bubba, Martin Burger, Tapio Helin, Luca Ratti

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Abstract

We consider a statistical inverse learning problem, where the task is to estimate a function f based on noisy point evaluations of Af, where A is a linear operator. The function Af is evaluated at i.i.d. random design points u n, n = 1, …, N generated by an unknown general probability distribution. We consider Tikhonov regularization with general convex and p-homogeneous penalty functionals and derive concentration rates of the regularized solution to the ground truth measured in the symmetric Bregman distance induced by the penalty functional. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.

Original languageEnglish
Pages (from-to)1193-1225
Number of pages33
JournalInverse Problems and Imaging
Volume17
Issue number6
Early online date1 Apr 2023
DOIs
Publication statusPublished - 1 Dec 2023

Bibliographical note

Funding Information:
TAB was supported by the Academy of Finland through the postdoctoral grant decision number 330522 and by the Royal Society through the Newton International Fellowship grant n. NIF\R1\201695. TAB and LR acknowledge support by the Academy of Finland through the Finnish Centre of Excellence in Inverse Modelling and Imaging 2018-2025, decision number 312339. Also INdAMGNCS and INdAM-GNAMPA are acknowledged. The work of MB has been supported by ERC via Grant EU FP7 ERC Consolidator Grant 615216 LifeInverse, by the German Ministry of Science and Technology (BMBF) under grant 05M2020-Deleto, and by the EU under grant 2020 NoMADS-DLV-777826. TH was supported by the Academy of Finland through decision number 326961. LR was supported by the Air Force Office of Scientific Research under award number FA8655-20-1-7027, and acknowledges the support of Fondazione Compagnia di San Paolo.

Funding Information:
Acknowledgments. TAB was supported by the Academy of Finland through the postdoctoral grant decision number 330522 and by the Royal Society through the Newton International Fellowship grant n. NIF\R1\201695. TAB and LR acknowledge support by the Academy of Finland through the Finnish Centre of Excellence in Inverse Modelling and Imaging 2018-2025, decision number 312339. Also INdAM-GNCS and INdAM-GNAMPA are acknowledged. The work of MB has been supported by ERC via Grant EU FP7 ERC Consolidator Grant 615216 LifeInverse, by the German Ministry of Science and Technology (BMBF) under grant 05M2020 - Deleto, and by the EU under grant 2020 NoMADS - DLV-777826. TH was supported by the Academy of Finland through decision number 326961. LR was supported by the Air Force Office of Scientific Research under award number FA8655-20-1-7027, and acknowledges the support of Fondazione Compagnia di San Paolo.

Keywords

  • Bregman distances
  • Variational regularization
  • computed tomography
  • error estimates
  • statistical learning

ASJC Scopus subject areas

  • Control and Optimization
  • Analysis
  • Discrete Mathematics and Combinatorics
  • Modelling and Simulation

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