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    Area of Science:

    • Machine Learning
    • Graph Theory
    • Data Mining

    Background:

    • Existing graph construction methods lack theoretical interpretation.
    • Previous approaches heuristically assigned graph weights.
    • A need exists for robust graph construction with distance preservation.

    Purpose of the Study:

    • Provide theoretical guarantees for distance-preserving graph construction.
    • Offer practical guidance on hyperparameter selection for graph sparsity.
    • Establish a clear interpretation of graph weights based on optimality conditions.

    Main Methods:

    • Estimating a density function of latent variables to retain pairwise distances.
    • Interpreting graph weights using optimality conditions and neighborhood graphs.
    • Developing systematic hyperparameter tuning strategies.

    Main Results:

    • Theoretical interpretation of graph construction is established.
    • Systematic methods for hyperparameter setting are provided.
    • Extensions include varied measure functions, graph refinement, and computational cost reduction.

    Conclusions:

    • The proposed graph construction method offers theoretical rigor and practical utility.
    • Experimental results validate theoretical findings and demonstrate competitive performance in semisupervised learning.
    • The work bridges the gap between heuristic graph construction and theoretical understanding.