Convergence analysis of equilibrium methods for inverse problems
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909798759399424 |
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| author | Obmann, Daniel Hwang, Gyeongha Haltmeier, Markus |
| author_facet | Obmann, Daniel Hwang, Gyeongha Haltmeier, Markus |
| contents | Solving inverse problems \(Ax = y\) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of \( \|A(\cdot) - y^δ\|^2 / 2 + α\mathcal{R}(\cdot)\), with \(\mathcal{R}\) a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of \(A^*(A x - y^δ) + αR(x) = 0\) for some regularization operator \(R\). When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, \(R\) is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_01421 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence analysis of equilibrium methods for inverse problems Obmann, Daniel Hwang, Gyeongha Haltmeier, Markus Numerical Analysis Computer Vision and Pattern Recognition Solving inverse problems \(Ax = y\) is central to a variety of practically important fields such as medical imaging, remote sensing, and non-destructive testing. The most successful and theoretically best-understood method is convex variational regularization, where approximate but stable solutions are defined as minimizers of \( \|A(\cdot) - y^δ\|^2 / 2 + α\mathcal{R}(\cdot)\), with \(\mathcal{R}\) a regularization functional. Recent methods such as deep equilibrium models and plug-and-play approaches, however, go beyond variational regularization. Motivated by these innovations, we introduce implicit non-variational (INV) regularization, where approximate solutions are defined as solutions of \(A^*(A x - y^δ) + αR(x) = 0\) for some regularization operator \(R\). When the regularization operator is the gradient of a functional, INV reduces to classical variational regularization. However, in methods like DEQ and PnP, \(R\) is not a gradient field, and the existing theoretical foundation remains incomplete. To address this, we establish stability and convergence results in this broader setting, including convergence rates and stability estimates measured via a absolute Bregman distance. |
| title | Convergence analysis of equilibrium methods for inverse problems |
| topic | Numerical Analysis Computer Vision and Pattern Recognition |
| url | https://arxiv.org/abs/2306.01421 |