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    This study introduces a new numerical integration framework for kernel machines. It uses symmetric interpolatory rules for accurate and efficient kernel approximation, improving large-scale machine learning performance.

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    Area of Science:

    • Machine Learning
    • Numerical Analysis

    Background:

    • Kernel machines are powerful tools in machine learning.
    • Approximating kernels is crucial for large-scale applications.
    • Existing methods face challenges in efficiency and accuracy.

    Purpose of the Study:

    • To develop an efficient quadrature framework for large-scale kernel machines.
    • To leverage numerical integration for accurate kernel approximation.
    • To unify existing quadrature-based kernel approximation methods.

    Main Methods:

    • Developed a quadrature framework using numerical integration.
    • Applied deterministic fully symmetric interpolatory rules for kernel approximation.
    • Introduced randomized rules via Monte-Carlo sampling and control variates.

    Main Results:

    • Reduced the number of nodes needed for kernel approximation while maintaining accuracy.
    • Achieved flexible feature mapping dimensions and controlled approximation discrepancies.
    • Demonstrated unbiasedness and variance reduction in stochastic rules.
    • Unified deterministic/stochastic interpolatory rules with existing methods.

    Conclusions:

    • The proposed framework offers an efficient and accurate approach to kernel approximation.
    • The methods compare favorably against existing representative techniques.
    • This work provides a unified perspective on quadrature-based kernel approximation.