On Optimal Regularization Parameters via Bilevel Learning

• URL: http://arxiv.org/abs/2305.18394v5
• Date: Mon, 22 Jan 2024 10:44:50 GMT
• Title: On Optimal Regularization Parameters via Bilevel Learning
• Authors: Matthias J. Ehrhardt, Silvia Gazzola and Sebastian J. Scott (Department of Mathematical Sciences, University of Bath, Bath, UK)
• Abstract summary: We provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems.
• Score: 0.06213771671016098
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Classical strategies used to determine a suitable parameter value include the discrepancy principle and the L-curve criterion, and in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters and involves solving a nested optimization problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still an open question. In particular, a necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems.

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