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The Kolmogorov-Arnold representation theorem revisited.
1University of Twente and Leiden University, Drienerlolaan 5, 7522 NB Enschede, The Netherlands.
Summary
The Kolmogorov-Arnold representation theorem, debated for explaining deep neural networks, can be modified. These modifications suggest deep networks, not just two-layer ones, naturally represent functions.
Area of Science:
- Artificial Intelligence
- Machine Learning
- Deep Learning Theory
Background:
- The Kolmogorov-Arnold representation theorem (KART) has been debated for its applicability to explaining deep neural networks.
- KART decomposes multivariate functions into inner and outer functions, mirroring a two-hidden-layer neural network structure.
- A key limitation is the outer function's potential for wild variation, even with smooth target functions.
Purpose of the Study:
- To investigate the relationship between the Kolmogorov-Arnold representation theorem and deep neural networks.
- To derive modifications of KART that address limitations regarding function smoothness.
- To explore how these modifications can be approximated by Rectified Linear Unit (ReLU) networks.
Main Methods:
- Deriving modified versions of the Kolmogorov-Arnold representation theorem.
- Analyzing the transfer of smoothness properties from the represented function to the outer function.
- Investigating approximations using ReLU networks.
Main Results:
- Modified KART versions successfully transfer smoothness properties to the outer function.
- These modified representations are well-approximated by ReLU networks.
- The findings suggest KART is more naturally interpreted as a deep neural network architecture.
Conclusions:
- The Kolmogorov-Arnold representation theorem, with modifications, supports the use of deep neural networks.
- Deep networks, rather than just two-layer networks, provide a more natural framework for KART.
- This work offers theoretical insights into the representational power of deep learning models.
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