Convex regularization in statistical inverse learning problems
This work addresses statistical inverse learning problems, providing theoretical guarantees for regularization methods, but it is incremental as it builds on existing Tikhonov regularization frameworks.
The paper tackles the problem of estimating a function from noisy evaluations of its linear transformation, using Tikhonov regularization with convex penalties, and derives concentration rates for the solution's convergence to the true function, with numerical validation in X-ray tomography.
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.