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    This study introduces deterministic sampling for matrix completion, moving beyond random data assumptions. New conditions ensure data recovery, with a novel method outperforming traditional approaches.

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    Area of Science:

    • * Data Science
    • * Applied Mathematics
    • * Signal Processing

    Background:

    • * Traditional matrix completion relies on random sampling assumptions, which fail in many real-world applications like data forecasting.
    • * Non-degenerate distributions of missing data challenge existing probabilistic models.
    • * There's a need for robust matrix completion methods that handle deterministic sampling.

    Purpose of the Study:

    • * To investigate real-valued matrix completion under deterministic sampling.
    • * To establish theoretical conditions for guaranteed matrix recovery from partial data.
    • * To develop and analyze novel algorithms for deterministic matrix completion.

    Main Methods:

    • * Proposing two novel conditions: isomeric condition and relative well-conditionedness.
    • * Proving these conditions are weaker than uniform sampling and that the isomeric condition is necessary for identifiability.
    • * Developing and analyzing the isomeric dictionary pursuit (IsoDP) method based on Schatten quasi-norm.

    Main Results:

    • * Demonstrated that the proposed conditions for deterministic sampling are weaker than uniform sampling assumptions.
    • * Proved the necessity of the isomeric condition for the identifiability of matrix completions.
    • * Showcased that the isomeric dictionary pursuit (IsoDP) method exhibits unique behaviors compared to traditional bilinear programs.

    Conclusions:

    • * The proposed isomeric and relative well-conditionedness conditions provide a theoretical foundation for deterministic matrix completion.
    • * The isomeric dictionary pursuit (IsoDP) offers a promising new direction for handling missing data in non-random sampling scenarios.
    • * This work advances the field of matrix completion by addressing limitations of random sampling assumptions.