Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems
This work provides theoretical guarantees for a widely used regularization method in inverse problems, but it is incremental as it extends existing analysis to non-linear settings.
The authors tackled the problem of reconstructing an estimator for non-linear statistical inverse learning problems using Tikhonov regularization, and they derived optimal convergence rates uniformly over solution classes defined by source conditions.
We study a non-linear statistical inverse learning problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization, MOR) approach to reconstruct the estimator of the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the ansatz of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.