Convex regularization in statistical inverse learning problems

• URL: http://arxiv.org/abs/2102.09526v2
• Date: Fri, 19 Feb 2021 07:29:17 GMT
• Title: Convex regularization in statistical inverse learning problems
• Authors: Tatiana A. Bubba and Martin Burger and Tapio Helin and Luca Ratti
• Abstract summary: We consider Tikhonov regularization with general convex and $p$-homogeneous penalty functionals. We derive concrete rates for Besov norm penalties and numerically demonstrate the correspondence with the observed rates in the context of X-ray tomography.
• Score: 1.7778609937758323
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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