Convex regularization in statistical inverse learning problems

TL;DR

This paper analyzes the statistical inverse learning problem, deriving convergence rates for convex regularization methods with various penalties, and demonstrates their effectiveness in X-ray tomography.

Contribution

It provides new theoretical convergence rates for convex regularization in inverse problems with general penalties, including Besov norms.

Findings

Derived concrete convergence rates for Besov norm penalties.

Numerical validation in X-ray tomography confirms theoretical predictions.

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.