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On the Convergence of Learning-Based Iterative Methods for Nonconvex Inverse Problems.

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    This study introduces the Flexible Iterative Modularization Algorithm (FIMA) for ill-posed inverse problems. FIMA offers a provable, globally convergent framework for learning-based iterative methods, improving upon current empirical approaches.

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    Area of Science:

    • Statistics
    • Machine Learning
    • Computer Vision
    • Applied Mathematics

    Background:

    • Ill-posed inverse problems are common in statistics, machine learning, and vision.
    • Learning-based iterative methods show empirical promise but lack theoretical rigor and flexible integration.
    • Current methods rely on intuition, hindering systematic development and analysis.

    Purpose of the Study:

    • To propose a generic and provable paradigm, the Flexible Iterative Modularization Algorithm (FIMA), for nonconvex inverse problems.
    • To provide theoretical guarantees for the convergence of learning-based iterative methods.
    • To enhance classical numerical methods in nonconvex scenarios.

    Main Methods:

    • Development of the Flexible Iterative Modularization Algorithm (FIMA).
    • Theoretical analysis of convergence properties for nonconvex inverse problems.
    • Design of flexible module scheduling policies.

    Main Results:

    • FIMA enables the generation of globally convergent trajectories for learning-based iterative methods.
    • Theoretical analysis confirms the provable convergence of the proposed framework.
    • Experimental validation demonstrates FIMA's superiority on real-world applications.

    Conclusions:

    • FIMA provides a robust and theoretically sound framework for addressing nonconvex inverse problems.
    • The modular and flexible nature of FIMA facilitates integration and analysis of iterative methods.
    • The proposed approach advances both learning-based and classical numerical techniques.