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Learned regularization for inverse problems

  • Martin Burger und Samira Kabri
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Data-driven Models in Inverse Problems
Ein Kapitel aus dem Buch Data-driven Models in Inverse Problems

Abstract

In this chapter, we provide a theoretically founded investigation of state-ofthe- art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization methods and their convergence in terms of the underlying data distributions, which paves the way for future theoretical studies. Based on a simple spectral learning model previously introduced for supervised learning, we investigate some key properties of different learning paradigms for inverse problems, which can be formulated independently of specific architectures. In particular, we investigate the regularization properties, bias, and critical dependence on training data distributions. Moreover, our framework allows highlighting and comparing the specific behavior of the different paradigms in the infinite-dimensional limit.

Abstract

In this chapter, we provide a theoretically founded investigation of state-ofthe- art learning approaches for inverse problems from the point of view of spectral reconstruction operators. We give an extended definition of regularization methods and their convergence in terms of the underlying data distributions, which paves the way for future theoretical studies. Based on a simple spectral learning model previously introduced for supervised learning, we investigate some key properties of different learning paradigms for inverse problems, which can be formulated independently of specific architectures. In particular, we investigate the regularization properties, bias, and critical dependence on training data distributions. Moreover, our framework allows highlighting and comparing the specific behavior of the different paradigms in the infinite-dimensional limit.

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/10.1515/9783111251233-002

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/10.1515/9783111251233-002/html?lang=de
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