A Bilevel Approach for Parameter Learning in Inverse Problems

TL;DR

This paper presents a bilevel optimization framework for learning regularization parameters in multi-penalty Tikhonov regularization, including theoretical conditions for solution existence and numerical validation.

Contribution

It introduces a bilevel approach for parameter learning in inverse problems, with new existence conditions and analysis of non-convex challenges.

Findings

Existence conditions for bilevel solutions are established.

Numerical experiments validate the theoretical results.

Discussion on non-convexity issues in the lower level problems.

Abstract

A learning approach to selecting regularization parameters in multi-penalty Tikhonov regularization is investigated. It leads to a bilevel optimization problem, where the lower level problem is a Tikhonov regularized problem parameterized in the regularization parameters. Conditions which ensure the existence of solutions to the bilevel optimization problem of interest are derived, and these conditions are verified for two relevant examples. Difficulties arising from the possible lack of convexity of the lower level problems are discussed. Optimality conditions are given provided that a reasonable constraint qualification holds. Finally, results from numerical experiments used to test the developed theory are presented.