Feasibility-based fixed point networks
Abstract Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-11-01
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| Series: | Fixed Point Theory and Algorithms for Sciences and Engineering |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13663-021-00706-3 |
| _version_ | 1818695247251636224 |
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| author | Howard Heaton Samy Wu Fung Aviv Gibali Wotao Yin |
| author_facet | Howard Heaton Samy Wu Fung Aviv Gibali Wotao Yin |
| author_sort | Howard Heaton |
| collection | DOAJ |
| description | Abstract Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling. Codes are available on Github: github.com/howardheaton/feasibility_fixed_point_networks . |
| first_indexed | 2024-12-17T13:42:26Z |
| format | Article |
| id | doaj.art-44f7425af370421a908ad4da0f3a81fc |
| institution | Directory Open Access Journal |
| issn | 2730-5422 |
| language | English |
| last_indexed | 2024-12-17T13:42:26Z |
| publishDate | 2021-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Algorithms for Sciences and Engineering |
| spelling | doaj.art-44f7425af370421a908ad4da0f3a81fc2022-12-21T21:46:15ZengSpringerOpenFixed Point Theory and Algorithms for Sciences and Engineering2730-54222021-11-012021111910.1186/s13663-021-00706-3Feasibility-based fixed point networksHoward Heaton0Samy Wu Fung1Aviv Gibali2Wotao Yin3Department of Mathematics, University of California, Los AngelesDepartment of Applied Mathematics and Statistics, Colorado School of MinesDepartment of Mathematics, ORT Braude CollegeDepartment of Mathematics, University of California, Los AngelesAbstract Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems; however, additional regularization is typically necessary to ensure accurate and stable recovery with respect to data perturbations. Hand-chosen analytic regularization can yield desirable theoretical guarantees, but such approaches have limited effectiveness recovering signals due to their inability to leverage large amounts of available data. To this end, this work fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs). Each F-FPN defines a collection of nonexpansive operators, each of which is the composition of a projection-based operator and a data-driven regularization operator. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction and a comparable neural network based on algorithm unrolling. Codes are available on Github: github.com/howardheaton/feasibility_fixed_point_networks .https://doi.org/10.1186/s13663-021-00706-3Convex feasibility problemProjectionAveragedFixed point networkNonexpansiveLearned regularizer |
| spellingShingle | Howard Heaton Samy Wu Fung Aviv Gibali Wotao Yin Feasibility-based fixed point networks Fixed Point Theory and Algorithms for Sciences and Engineering Convex feasibility problem Projection Averaged Fixed point network Nonexpansive Learned regularizer |
| title | Feasibility-based fixed point networks |
| title_full | Feasibility-based fixed point networks |
| title_fullStr | Feasibility-based fixed point networks |
| title_full_unstemmed | Feasibility-based fixed point networks |
| title_short | Feasibility-based fixed point networks |
| title_sort | feasibility based fixed point networks |
| topic | Convex feasibility problem Projection Averaged Fixed point network Nonexpansive Learned regularizer |
| url | https://doi.org/10.1186/s13663-021-00706-3 |
| work_keys_str_mv | AT howardheaton feasibilitybasedfixedpointnetworks AT samywufung feasibilitybasedfixedpointnetworks AT avivgibali feasibilitybasedfixedpointnetworks AT wotaoyin feasibilitybasedfixedpointnetworks |