Error bounds for deep ReLU networks using the Kolmogorov-Arnold superposition theorem.

Area of Science:

• Multivariate calculus
• Approximation theory
• Deep learning theory

Background:

• The curse of dimensionality poses significant challenges in approximating complex multivariate functions.
• Traditional approximation methods struggle with high-dimensional data.
• Deep Rectified Linear Unit (ReLU) networks offer potential solutions but require theoretical grounding.

Purpose of the Study:

• To develop a theoretical framework for approximating multivariate functions using deep ReLU networks.
• To demonstrate how deep ReLU networks can overcome the curse of dimensionality.
• To provide a constructive proof related to the Kolmogorov-Arnold superposition theorem.

Main Methods:

• Leveraging a constructive proof of the Kolmogorov-Arnold superposition theorem.
• Identifying a specific subset of continuous multivariate functions.
• Analyzing the efficient approximation capabilities of deep ReLU networks for outer superposition functions.

Main Results:

• A novel theorem concerning the approximation of multivariate functions by deep ReLU networks.
• Demonstration of reduced impact from the curse of dimensionality.
• Efficient approximation of outer superposition functions for a specific function class.

Conclusions:

• Deep ReLU networks offer a promising approach to efficiently approximate multivariate functions.
• The theoretical framework supports the use of deep ReLU networks in high-dimensional spaces.
• The findings contribute to a deeper understanding of the capabilities of deep learning models.