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    Area of Science:

    • Data Science
    • Machine Learning
    • Computer Vision

    Background:

    • Traditional matrix completion (MC) and tensor completion (TC) assume random missing data, failing with real-world structural missingness where data is concentrated.
    • Structural missingness, common in real-world datasets, significantly degrades the performance of existing MC and TC techniques.

    Purpose of the Study:

    • To introduce a novel framework, multi-matrix completion (MMC), designed to effectively handle structural missingness in data reconstruction.
    • To develop a method that simultaneously reconstructs multiple matrices, capturing inter-matrix correlations for improved completion accuracy.

    Main Methods:

    • The proposed multi-matrix completion (MMC) framework utilizes tri-factorization across matrices to leverage correlations between them.
    • Tikhonov regularization is applied to individual matrices within the MMC framework to exploit their inherent correlations.
    • The optimization problem is solved using the alternating direction method of multipliers, with theoretical guarantees for global convergence.

    Main Results:

    • Multi-matrix completion (MMC) demonstrates superior performance in restoring data with structural missingness compared to traditional MC and TC methods.
    • Experiments in image and video restoration validate the versatility and effectiveness of the MMC framework.
    • MMC is robust to data smoothness variations, offering broader applicability than Fourier transform-based tensor completion methods.

    Conclusions:

    • The novel multi-matrix completion (MMC) framework effectively addresses the challenge of structural missingness in data reconstruction.
    • MMC offers a versatile and high-performing solution for various data restoration tasks, including image and video processing.
    • The method's ability to handle both random and structural missingness, along with its theoretical convergence guarantees, marks a significant advancement in completion techniques.