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Smooth Kolmogorov Arnold networks enabling structural knowledge representation

Moein E. Samadi, Younes Müller, Andreas Schuppert

TL;DR

This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes, thereby enhancing model reliability and performance in computational biomedicine.

Abstract

Kolmogorov-Arnold Networks (KANs) offer an efficient and interpretable alternative to traditional multi-layer perceptron (MLP) architectures due to their finite network topology. However, according to the results of Kolmogorov and Vitushkin, the representation of generic smooth functions by KAN implementations using analytic functions constrained to a finite number of cutoff points cannot be exact. Hence, the convergence of KAN throughout the training process may be limited. This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes. By leveraging inherent structural knowledge, KANs may reduce the data required for training and mitigate the risk of generating hallucinated predictions, thereby enhancing model reliability and performance in computational biomedicine.

Smooth Kolmogorov Arnold networks enabling structural knowledge representation

TL;DR

This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes, thereby enhancing model reliability and performance in computational biomedicine.

Abstract

Kolmogorov-Arnold Networks (KANs) offer an efficient and interpretable alternative to traditional multi-layer perceptron (MLP) architectures due to their finite network topology. However, according to the results of Kolmogorov and Vitushkin, the representation of generic smooth functions by KAN implementations using analytic functions constrained to a finite number of cutoff points cannot be exact. Hence, the convergence of KAN throughout the training process may be limited. This paper explores the relevance of smoothness in KANs, proposing that smooth, structurally informed KANs can achieve equivalence to MLPs in specific function classes. By leveraging inherent structural knowledge, KANs may reduce the data required for training and mitigate the risk of generating hallucinated predictions, thereby enhancing model reliability and performance in computational biomedicine.
Paper Structure (4 sections, 3 equations, 1 figure)

This paper contains 4 sections, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Smoothness and finite nested functions
  3. Consequences of smoothness in finite networks of nested functions
  4. Conclusion

Figures (1)

  • Figure 1: Convergence of the validation RMSE of $w(u(x_1, x_2), v(y_1, y_2))$ for learning the target variables $z = x_1^2 x_2 + y_1 y_2^2$ and $z' = x_1 y_1 y_2 + x_1 x_2 y_2$ by strctured XGBoost regressor model. The model structure is well-suited for predicting $z$, as shown by the decreasing RMSE. However, it struggles to predict $z'$, indicating that $z'$ lies outside the representable function space of the model.