Convex regularization in statistical inverse learning problems

Tatiana Bubba; Martin Burger; Tapio Helin; Luca Ratti

doi:10.3934/ipi.2023013

Convex regularization in statistical inverse learning problems

Tatiana Bubba, Martin Burger, Tapio Helin, Luca Ratti

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

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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 language	English
Pages (from-to)	1193-1225
Number of pages	33
Journal	Inverse Problems and Imaging
Volume	17
Issue number	6
Early online date	1 Apr 2023
DOIs	https://doi.org/10.3934/ipi.2023013
Publication status	Published - 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

Access to Document

10.3934/ipi.2023013

ConvexRegu_StatInvProb
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Inverse Problems and Imaging following peer review. The definitive publisher-authenticated version Tatiana A. Bubba, Martin Burger, Tapio Helin, Luca Ratti. Convex regularization in statistical inverse learning problems. Inverse Problems and Imaging, 2023, 17(6): 1193-1225. doi: 10.3934/ipi.2023013 is available online at: https://www.aimsciences.org//article/doi/10.3934/ipi.2023013
Accepted author manuscript, 907 KBLicence: CC BY

https://arxiv.org/abs/2102.09526

Cite this

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title = "Convex regularization in statistical inverse learning problems",

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.",

keywords = "Bregman distances, Variational regularization, computed tomography, error estimates, statistical learning",

author = "Tatiana Bubba and Martin Burger and Tapio Helin and Luca Ratti",

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. ",

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AU - Burger, Martin

AU - Helin, Tapio

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N1 - 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.

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N2 - 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.

AB - 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.

KW - Bregman distances

KW - Variational regularization

KW - computed tomography

KW - error estimates

KW - statistical learning

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M3 - Article

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VL - 17

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EP - 1225

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

IS - 6

ER -

Convex regularization in statistical inverse learning problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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