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    Area of Science:

    • Data Science
    • Machine Learning
    • Signal Processing

    Background:

    • Matrix completion (MC) algorithms typically struggle with impulsive noise.
    • Existing outlier-resistant methods like entry-wise lp-norm and M-estimation have limitations, including optimal parameter selection and breakdown points.

    Purpose of the Study:

    • To develop robust matrix completion algorithms resistant to impulsive noise.
    • To introduce a novel approach using the entrywise l0-norm for anomaly separation and outlier detection.

    Main Methods:

    • Exploited the entrywise l0-norm to identify and separate outliers from observed matrices.
    • Utilized a Laplacian kernel for automated outlier detection and adaptive penalty parameter updates.
    • Developed block coordinate descent (BCD) and majorization-minimization (MM) based algorithms (l0-BCD, l0-BCD-F, l0-BCD-MM, l0-BCD-MM-F).

    Main Results:

    • The proposed l0-BCD algorithm guarantees convergence for outlier detection and separation.
    • l0-BCD-F offers higher computational efficiency by treating outliers as missing entries.
    • l0-BCD-MM and l0-BCD-MM-F effectively handle nonnegativity constraints with closed-form updates.

    Conclusions:

    • The novel algorithms significantly outperform state-of-the-art methods in both recovery accuracy and computational efficiency.
    • The entrywise l0-norm approach provides a robust solution for matrix completion in the presence of impulsive noise.
    • Demonstrated effectiveness in practical applications like image inpainting and hyperspectral image recovery.