A class of parameter choice rules for fractional Tikhonov regularization scheme in learning theory

Abstract

Klann and Ramlau [16] hypothesized fractional Tikhonov regularization as an interpolation between generalized inverse and Tikhonov regularization. In fact, fractional schemes can be viewed as a generalization of the Tikhonov scheme. One of the motives of this work is the major pitfall of the a priori parameter choice rule, which primarily relies on source conditions that are often unknown. It necessitates the need for advocating a data-driven approach (a posteriori choice strategy). We briefly overview fractional scheme in learning theory and propose a modified Engl type [9] discrepancy principle, thus integrating supervised learning into the field of inverse problems. In due course of the investigation, we effectively explored the relation between learning from examples and the inverse problems. We demonstrate the regularization properties and establish the convergence rate of this scheme. Finally, the theoretical results are corroborated using two well known examples in learning theory.