Startseite Mathematik On optimal regularization parameters via bilevel learning
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On optimal regularization parameters via bilevel learning

  • Matthias J. Ehrhardt , Silvia Gazzola und Sebastian J. Scott
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Data-driven Models in Inverse Problems
Ein Kapitel aus dem Buch Data-driven Models in Inverse Problems

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. Whereas 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 chapter, 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.

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. Whereas 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 chapter, 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.

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Part I: Mathematical aspects of data-driven methods in inverse problems
  5. On optimal regularization parameters via bilevel learning 1
  6. Learned regularization for inverse problems 39
  7. Inverse problems with learned forward operators 73
  8. Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
  9. Learned reconstruction methods for inverse problems: sample error estimates 163
  10. Statistical inverse learning problems with random observations 201
  11. General regularization in covariate shift adaptation 245
  12. Part II: Applications of data-driven methods in inverse problems
  13. Analysis of generalized iteratively regularized Landweber iterations driven by data 273
  14. Integration of model- and learning-based methods in image restoration 303
  15. Dynamic computerized tomography using inexact models and motion estimation 331
  16. Deep Bayesian inversion 359
  17. Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
  18. Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
  19. Classification with neural networks with quadratic decision functions 471
  20. Index 495
https://doi.org/10.1515/9783111251233-001

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. Part I: Mathematical aspects of data-driven methods in inverse problems
  5. On optimal regularization parameters via bilevel learning 1
  6. Learned regularization for inverse problems 39
  7. Inverse problems with learned forward operators 73
  8. Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
  9. Learned reconstruction methods for inverse problems: sample error estimates 163
  10. Statistical inverse learning problems with random observations 201
  11. General regularization in covariate shift adaptation 245
  12. Part II: Applications of data-driven methods in inverse problems
  13. Analysis of generalized iteratively regularized Landweber iterations driven by data 273
  14. Integration of model- and learning-based methods in image restoration 303
  15. Dynamic computerized tomography using inexact models and motion estimation 331
  16. Deep Bayesian inversion 359
  17. Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
  18. Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
  19. Classification with neural networks with quadratic decision functions 471
  20. Index 495
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Heruntergeladen am 12.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/9783111251233-001/html?lang=de
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