Convex regularization in statistical inverse learning problems

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 prob...

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Bibliographic Details
Published in:arXiv.org 2021-11
Main Authors: Bubba, Tatiana A, Burger, Martin, Helin, Tapio, Ratti, Luca
Format: Article
Language:English
Subjects:
Evaluation
Ground truth
Learning
Linear operators
Regularization
Statistical analysis
Online Access:Get full text
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Summary: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.
ISSN:2331-8422