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    Area of Science:

    • Artificial intelligence
    • Machine learning
    • Neural networks

    Background:

    • Artificial neural networks are inspired by biological systems, leading to diverse neuron designs.
    • Quadratic neurons, replacing inner-products with quadratic operations, show promise but their performance advantage is debated.

    Purpose of the Study:

    • To investigate whether the superior performance of quadratic neural networks stems from increased parameters or inherent expressive power.
    • To theoretically and empirically validate the parametric efficiency of quadratic networks.

    Main Methods:

    • Theoretical analysis of approximation efficiency in real space and manifolds.
    • Examination within the Barron space to compare approximation errors with conventional networks.
    • Empirical evaluation on synthetic, benchmark, and real-world datasets.

    Main Results:

    • Quadratic networks exhibit parametric efficiency, confirming their intrinsic expressive capability.
    • Quadratic neurons effectively model nonlinear interactions, a challenge for conventional neurons.
    • Approximation efficiency is superior in quadratic networks, especially in high-dimensional spaces.

    Conclusions:

    • The enhanced performance of quadratic neural networks is attributed to their intrinsic expressive capabilities.
    • Quadratic networks offer significant advantages in parametric efficiency, particularly for tasks involving complex nonlinearities.
    • Findings support the broader applicability and potential of quadratic networks in various machine learning domains.