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    This study introduces catalog networks, a type of neural network that overcomes the curse of dimensionality in high-dimensional approximation problems. These networks can be efficiently approximated by rectified linear unit networks, offering solutions without the curse of dimensionality.

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    Area of Science:

    • Machine Learning
    • Numerical Analysis
    • Computer Science

    Background:

    • The curse of dimensionality poses a significant challenge in high-dimensional approximation problems, limiting the efficiency of traditional methods.
    • Neural networks have shown promise in overcoming these limitations, but their application in high dimensions requires further investigation.
    • Understanding the approximation capabilities of neural networks for complex functions is crucial for advancing computational mathematics.

    Purpose of the Study:

    • To develop a theoretical framework demonstrating how neural networks can surmount the curse of dimensionality in approximation tasks.
    • To introduce and analyze 'catalog networks,' a generalized neural network architecture with adaptable activation functions.
    • To establish conditions under which catalog networks can be efficiently approximated by rectified linear unit (ReLU) networks.

    Main Methods:

    • The study proposes a framework based on 'catalog networks,' where activation functions can vary across layers from a predefined set.
    • Theoretical analysis is employed to establish the approximation capabilities of catalog networks.
    • The research investigates the efficient approximation of catalog networks using rectified linear unit (ReLU)-type networks, providing parameter estimates.

    Main Results:

    • Catalog networks are shown to be a rich family of continuous functions capable of overcoming the curse of dimensionality.
    • Under specific conditions on the function catalog, catalog networks can be efficiently approximated by ReLU networks.
    • Precise estimates for the number of parameters required for a given approximation accuracy are provided.

    Conclusions:

    • The developed framework confirms that neural networks, specifically catalog networks, can effectively address the curse of dimensionality.
    • Rectified linear unit networks can efficiently approximate catalog networks, leading to classes of functions that avoid the curse of dimensionality.
    • This research offers a theoretical foundation for designing neural network architectures that are scalable and efficient in high-dimensional spaces.