On the expressiveness and spectral bias of KANs (2410.01803v2)

Abstract: Kolmogorov-Arnold Networks (KAN) \cite{liu2024kan} were very recently proposed as a potential alternative to the prevalent architectural backbone of many deep learning models, the multi-layer perceptron (MLP). KANs have seen success in various tasks of AI for science, with their empirical efficiency and accuracy demostrated in function regression, PDE solving, and many more scientific problems. In this article, we revisit the comparison of KANs and MLPs, with emphasis on a theoretical perspective. On the one hand, we compare the representation and approximation capabilities of KANs and MLPs. We establish that MLPs can be represented using KANs of a comparable size. This shows that the approximation and representation capabilities of KANs are at least as good as MLPs. Conversely, we show that KANs can be represented using MLPs, but that in this representation the number of parameters increases by a factor of the KAN grid size. This suggests that KANs with a large grid size may be more efficient than MLPs at approximating certain functions. On the other hand, from the perspective of learning and optimization, we study the spectral bias of KANs compared with MLPs. We demonstrate that KANs are less biased toward low frequencies than MLPs. We highlight that the multi-level learning feature specific to KANs, i.e. grid extension of splines, improves the learning process for high-frequency components. Detailed comparisons with different choices of depth, width, and grid sizes of KANs are made, shedding some light on how to choose the hyperparameters in practice.

• The paper demonstrates that KANs match or exceed MLPs in representation power through efficient reparameterizations.
• It reveals that KANs exhibit reduced spectral bias, enabling balanced learning of both low- and high-frequency components.
• Empirical and theoretical analyses highlight KANs’ superior efficiency in tasks like function regression and solving PDEs.

On the expressiveness and spectral bias of KANs

The paper "On the expressiveness and spectral bias of KANs" provides an in-depth comparative analysis of Kolmogorov-Arnold Networks (KANs) and the more conventional Multi-Layer Perceptrons (MLPs) from both theoretical and empirical perspectives. Let's delve into the key contributions of this research and explore its implications.

Key Contributions

1. Representation and Approximation Power:
   • The paper provides a rigorous theoretical comparison between the representation capabilities of KANs and MLPs. The authors demonstrate that any MLP with ReLU$^k$ activation functions can be reparameterized into a KAN with a comparable number of parameters, establishing that KANs have at least the same representation power as MLPs.
   • Conversely, they also show that KANs can be represented using MLPs, although the number of parameters increases by a factor proportional to the KAN grid size. This asymmetry suggests potential efficiency advantages for KANs in representing certain functions, especially those requiring a large grid size.
2. Spectral Bias:
   • The spectral bias problem, where standard MLPs tend to learn low-frequency functions first, was analyzed in-depth. By studying the training dynamics of KANs, it was found that KANs are less biased towards low frequencies than MLPs. This is because KANs, due to their grid extension of splines, exhibit a more balanced learning process for high-frequency components.
   • The paper provides theoretical evidence showing that KANs with a large grid size offer better parameter efficiency and gradient descent dynamics that do not inherently prioritize low frequencies. These findings are supported by numerical experiments which show that KANs consistently display less spectral bias across a variety of tasks, such as 1D frequency fitting, high-dimensional Gaussian kernel fitting, and solving high-frequency Poisson equations.

Implications

Practical Implications

The reduced spectral bias in KANs has significant practical implications. In applications such as scientific computing, where high-frequency components are critical, the less biased learning dynamics of KANs could lead to superior performance. The practical utility of KANs was illustrated through their use in function regression and PDE solving, showcasing their efficiency and accuracy in various tasks. Their robustness to high-frequency components and multi-level learning strategies could drive advancements in fields such as computational physics and engineering simulations.

Theoretical Implications

The theoretical insights into the representation capabilities and spectral bias dynamics provided by this research complement the existing body of work on neural network approximation theories. The results regarding KANs' efficiency in approximating certain classes of functions naturally extend to Sobolev spaces, reinforcing the practical relevance of these networks in high-dimensional function approximation. Additionally, the demonstrated reduction in spectral bias theoretically underpins the empirical success of KANs in scientific applications.

Future Directions

Several avenues for future research are suggested by these findings:

• Deeper Theoretical Analysis:

Further investigation into deeper KAN architectures and their dynamics could provide more comprehensive theoretical foundations and practical guiding principles for their deployment in various computational tasks.

• Expanded Experimental Validation:

While the current paper focuses on fundamental tasks, further experimental work could explore more complex and diverse problem domains to fully establish the advantages and limitations of KANs in real-world applications.

• Hybrid Models:

Investigating hybrid models that combine KANs with other neural architectures or numerical methods could exploit the strengths of both, potentially leading to more powerful and flexible models.

• Hyperparameter Optimization:

In-depth studies on the optimal selection of KAN hyperparameters (such as depth, width, and grid size) tailored to specific tasks could enhance performance and streamline their application.

Conclusion

The comparative paper of KANs and MLPs in the context of expressiveness and spectral bias reveals critical insights that hold promise for enhancing neural network architectures in both theoretical and practical dimensions. The demonstrated efficiency and balanced frequency learning of KANs position them as a promising alternative in tasks requiring high accuracy and interpretability, particularly in scientific computing. This research lays a foundational understanding that could fuel future innovations and optimizations in neural network design and application.