Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems

Abstract

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.

References in corpus (3)

• Methodology and convergence rates for functional linear regression
• Convergence Analysis of (Statistical) Inverse Problems under Conditional Stability Estimates
• Optimal rates for the regularized learning algorithms under general source condition

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• Statistical Inverse Problems in Hilbert Scales
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• Sampling theorems for inverse problems on Riemannian manifolds