TY - JOUR

T1 - Shearlet-based regularization in statistical inverse learning with an application to X-ray tomography

AU - Bubba, Tatiana

AU - Ratti, Luca

N1 - Funding Information:
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme ‘Mathematics of Deep Learning’ where work on this paper was undertaken. This work was supported by EPSRC Grant No. EP/R014604/1. TAB is supported by the Royal Society through the Newton International Fellowship Grant No. NIF∖R1∖201695 and was partially supported by the Academy of Finland through the postdoctoral Grant, decision Number 330522. LR is supported by the Air Force Office of Scientific Research under Award Number FA8655-20-1-7027. Also INdAM-GNCS and INdAM-GNAMPA are acknowledged. The authors would also like to thank Tapio Helin and Martin Burger for introducing them to the fascinating field of statistical inverse learning problems.

PY - 2022/5/31

Y1 - 2022/5/31

N2 - Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, p-homogeneous regularizers with p (1, 2], in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with .," p -norm regularization, with p (1, 2), for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The p = 1 case is approached as the limit case (1, 2) p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from "-convergence theory. We numerically validate our theoretical results in the context of x-ray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearlet-based regularization.

AB - Statistical inverse learning theory, a field that lies at the intersection of inverse problems and statistical learning, has lately gained more and more attention. In an effort to steer this interplay more towards the variational regularization framework, convergence rates have recently been proved for a class of convex, p-homogeneous regularizers with p (1, 2], in the symmetric Bregman distance. Following this path, we take a further step towards the study of sparsity-promoting regularization and extend the aforementioned convergence rates to work with .," p -norm regularization, with p (1, 2), for a special class of non-tight Banach frames, called shearlets, and possibly constrained to some convex set. The p = 1 case is approached as the limit case (1, 2) p → 1, by complementing numerical evidence with a (partial) theoretical analysis, based on arguments from "-convergence theory. We numerically validate our theoretical results in the context of x-ray tomography, under random sampling of the imaging angles, using both simulated and measured data. This application allows to effectively verify the theoretical decay, in addition to providing a motivation for the extension to shearlet-based regularization.

KW - Bregman distance

KW - convergence rates

KW - convex regularization

KW - shearlets

KW - statistical learning

KW - wavelets

KW - x-ray tomography

UR - http://www.scopus.com/inward/record.url?scp=85128759719&partnerID=8YFLogxK

U2 - 10.1088/1361-6420/ac59c2

DO - 10.1088/1361-6420/ac59c2

M3 - Article

VL - 38

SP - 1

EP - 43

JO - Inverse Problems

JF - Inverse Problems

SN - 0266-5611

IS - 5

M1 - 054001

ER -